Let H be a group and . Then for a group G and .
This is true if H is a cyclic group. I want to know whether is this true for any other finite groups.
i don't know if you understood the idea in my previous post or not. as long as has a normal subgroup which is not characteristic, the claim in your problem would be false.
here's a simple example of finite abelian groups: let define by: then let and and define
by: clearly and