Let $G = A \rtimes B,$ where $A$ is a finite $p$-group and $B$ is a finite $p'$ group. When can we say that $Z(G) \subseteq A?$
For example the center of the group $C_3 \rtimes C_4$ is $C_2$ and hence $Z(G)$ is not a subgroup of $C_3.$ Thus in this case $Z(G)$ is not contained in $A.$ I wanted to ask that under what conditions on $A$ and $B,$ does this hold true. Is there some nice reference where I can read more details about the general theory behind it.
Writing $G= A \rtimes B$ does not define $G$ uniquely. For example, you cannot talk about "the group $C_3 \rtimes C_4$", because there are two groups with that structure, and their centres are $C_2$ and $C_4$.
To define $G = A \rtimes_\phi B$, you need to specify a homomorphism $\phi: B \to {\rm Aut}(A)$. Then the condition for $Z(G) \le A$ is that $\ker \phi$ is trivial.