Group extension

Question

I am getting blurred about group extensions. Let $A,B,C$ be groups.

1. If $G=(A{:}B).C$ and $A$ is characteristic in $G$, then $G=A.(B.C)$. But is it also true that $G=A{:}(B.C)$ ?

2. If $G=A.(B{:}C)$, then $G=(A.B).C$. But is it true that $G=(A.B){:}C$ ?

Note that $G=A.B$ means $G/A\cong B$; and $G=A{:}B$ means semidirect product of $A$ and $B$ with $A\lhd G$.

Answer 1

As to (1), no. Take $G$ to be the direct product of a cyclic group of order $9$ by a group of order $2$. Let $A$ be the unique subgroup of order $3$, and $B$ the unique subgroup of order $2$, so $C$ is cyclic of order $3$. Then $AB$ is the unique subgroup of order $6$, so $A$ cannot have a complement $B.C$ of order $6$.

The very same example show that the answer to (2) is also negative, as there's no subgroup $C$ of order $3$ besides $A$.