I am confused about the following situation: suppose that $G$ is a cyclic group and that have an exact sequence $$ 1 \to N \to G \to H \to 1$$ where $N=C_n$, and $H\simeq C_k$. So if $G$ is generated by $x$, then $N$ is generated by $x^k$.
Suppose that there is an action of $N$ on a vector space $V$ of dimension $v$, specified by a certain $v \times v$ matrix $A$. Moreover, suppose that we also have an action of $H$ on $V$, given by another $v \times v$ matrix $B$.
I am interested in the following questions:
I can inflate $B$ to a representation of $G$, and consider the product $AB$. Does this give me a representation of $G$? What conditions do I need, if any?
The opposite direction: given a representation of $G$, when can I find $A$ and $B$ as above such that the construction in 1. gives me a representation of $G$?