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Try to share the Root SharePoint Site to Group A

• As Limited Access to Can view specific lists, document libraries, list items, folders, or documents when given permissions.
• And as contribute to the SharePoint list that has the workflow with its related Task List and History list.

You may also need to check "How to elevate Workflow permissions in SharePoint 2013" by doing the following :

• Allow workflow to use app permissions.
• Grant full control permission to a workflow.
• Develop the workflow to wrap actions inside an App Step using SharePoint Designer.

For more details about How to apply the above steps check The Workflow was Suspended with Unauthorized HTTP / elevate Workflow permissions in SharePoint 2013

Update :

• The behaviour is by design. A user at least needs to have contributed permissions on the root site, in order to launch workflows manually as mention at this thread Manually start SharePoint 2010/2013 workflow in SharePoint 2013 farm?

• Also, check the latest CU that has been installed , it's preferred to apply Service Pack 1 if it's not applied where maybe this issue solved at it. hope it helped you

Try to share the Root SharePoint Site to Group A

• As Limited Access to Can view specific lists, document libraries, list items, folders, or documents when given permissions.
• And as contribute to the SharePoint list that has the workflow with its related Task List and History list.

You may also need to check "How to elevate Workflow permissions in SharePoint 2013" by doing the following :

• Allow workflow to use app permissions.
• Grant full control permission to a workflow.
• Develop the workflow to wrap actions inside an App Step using SharePoint Designer.

For more details about How to apply the above steps check The Workflow was Suspended with Unauthorized HTTP / elevate Workflow permissions in SharePoint 2013

Update :

• The behaviour is by design. A user at least needs to have contributed permissions on the root site, in order to launch workflows manually as mention at this thread Manually start SharePoint 2010/2013 workflow in SharePoint 2013 farm?

• Also, check the latest CU that has been installed , it's preferred to apply Service Pack 1 if it's not applied where maybe this issue solved at it. hope it helped you

Try to share the Root SharePoint Site to group A

• As Limited Access to Can view specific lists, document libraries, list items, folders, or documents when given permissions.
• And as contribute to the SharePoint list that has the workflow with its related Task List and History list.

You may also need to check how to elevate Workflow permissions in SharePoint 2013 by doing the following :

• Allow workflow to use app permissions.
• Grant full control permission to a workflow.
• Develop the workflow to wrap actions inside an App Step using SharePoint Designer.

For more details about How to apply the above steps check The Workflow was Suspended with Unauthorized HTTP / elevate Workflow permissions in SharePoint 2013

Update :

• The behaviour is by design. A user at least needs to have contributed permissions on the root site, in order to launch workflows manually as mention at this thread Manually start SharePoint 2010/2013 workflow in SharePoint 2013 farm?

• Also, check the latest CU that has been installed , it's preferred to apply Service Pack 1 if it's not applied where maybe this issue solved at it. hope it helped you

Try to share the Root SharePoint Site to Group A

• As Limited Access to Can view specific lists, document libraries, list items, folders, or documents when given permissions.
• And as contribute to the SharePoint list that has the workflow with its related Task List and History list.

You may also need to check "How to elevate Workflow permissions in SharePoint 2013" by doing the following :

• Allow workflow to use app permissions.
• Grant full control permission to a workflow.
• Develop the workflow to wrap actions inside an App Step using SharePoint Designer.

For more details about How to apply the above steps check The Workflow was Suspended with Unauthorized HTTP / elevate Workflow permissions in SharePoint 2013

Update :

• The behaviour is by design. A user at least needs to have contributed permissions on the root site, in order to launch workflows manually as mention at this thread Manually start SharePoint 2010/2013 workflow in SharePoint 2013 farm?

• Also, check the latest CU that has been installed , it's preferred to apply Service Pack 1 if it's not applied where maybe this issue solved at it. hope it helped you

Try to share the Root SharePoint Site to group A

• As Limited Access to Can view specific lists, document libraries, list items, folders, or documents when given permissions.
• And as contribute to the SharePoint list that has the workflow with its related Task List and History list.

You may also need to check how to elevate Workflow permissions in SharePoint 2013 by doing the following :

• Allow workflow to use app permissions.
• Grant full control permission to a workflow.
• Develop the workflow to wrap actions inside an App Step using SharePoint Designer.

For more details about How to apply the above steps check The Workflow was Suspended with Unauthorized HTTP / elevate Workflow permissions in SharePoint 2013

Update : I think you need to apply CU Sept as mention at this thread Manually start SharePoint 2010/2013 workflow in SharePoint 2013 farm? hope it helped you

Try to share the Root SharePoint Site to group A

• As Limited Access to Can view specific lists, document libraries, list items, folders, or documents when given permissions.
• And as contribute to the SharePoint list that has the workflow with its related Task List and History list.

You may also need to check how to elevate Workflow permissions in SharePoint 2013 by doing the following :

• Allow workflow to use app permissions.
• Grant full control permission to a workflow.
• Develop the workflow to wrap actions inside an App Step using SharePoint Designer.

For more details about How to apply the above steps check The Workflow was Suspended with Unauthorized HTTP / elevate Workflow permissions in SharePoint 2013

Update :

• The behaviour is by design. A user at least needs to have contributed permissions on the root site, in order to launch workflows manually as mention at this thread Manually start SharePoint 2010/2013 workflow in SharePoint 2013 farm?

• Also, check the latest CU that has been installed , it's preferred to apply Service Pack 1 if it's not applied where maybe this issue solved at it. hope it helped you