Action on $G$ by inner automorphism

Question

I wonder something about an action of a group $A$ on a group $G$ by a automorphism;

There are many nice result related with some restrictions such as when $(|A|,|G|)=1$ , $G$ is abelian or $[G,A,A]=e$...

I wonder can we say anything when we know $A$ act on $G$ by inner automorphism i.e.for any $a\in A$ and $g\in G$ there exist an $h\in G$ s.t. $g^a=g^h$.

Answer 1

There is nothing you can say, because any group $A$ has the trivial group action on $G$, and in this case $A$ acts by inner automorphisms.

The assumption that $A$ acts on $G$ by inner automorphisms just gives you some homomorphism $f: A \rightarrow \operatorname{Inn}(G)$. Without any restriction on this action or on the groups $A$ and $G$, you cannot deduce anything from this.