Let $G = A \rtimes B$ be a finite group ($A$ is the normal subgroup), where $A$ is a non-abelian $p$-group and $(|B|,p) = 1.$
Then what can we say about the center of $G?$ Does it always lie inside $A?$
What should be the condition on $B$ so that $Z(G) \subseteq A?$ Can we conclude anything if $|B| < |A|$
The only fact that I know is that in some cases if $A$ is abelian (but $G$ is non-abelian), then $C_A(B) = Z(G).$ But I have no idea what happens if $A$ is non-abelian.