qgis - Copying attributes from one polygon layer to another?

I have a problem that I can't seem to get my head around. I have two polygon layers:

• Polygon A - is a subset of polygon B with the same fields and has identical polygons to Polygon B


• Polygon B - has the attribute data I want to be in Polygon A

How can this be done? I tried the QGIS tool "Join Attributes by Location" but as some of the polygons are within others, it tends to link to the first intersect it finds (the outer polygon).