I have the following problem. Let $$ 0\to A\to G\to Q\to 1 $$ be a central group extension with $A$ abelian. Assume that this extension splits, i.e., $G\cong A\rtimes Q$.
Now consider an action of another group $H$ on $G$ in such a way, that $H$ fixes $A$. In this case, it $H$ acts also on $Q$ in a way 'coherent' with the extension above. Under this assumption there is another extension of the form: $$ 0\to A\to G\rtimes H\to Q\rtimes H\to 1. $$
Now the question I have is: since the first extension is split, does it follow that the second extension is? In other words, do we have that $(A\rtimes Q)\rtimes H\cong A\rtimes(Q\rtimes H)$?
I am pretty sure this is not true, so I would be very grateful for some nice counterexample!