Let $D_{2^n}$ be a dihedal group of order $2^n$ and $Q_{2^n}$ be a quaternion group of order $2^n.$
I am looking for a reference about the structure of $Aut(D_{2^n}*C_{2^k})$ and $Aut(Q_{2^n}*C_{2^k})$ where $*$ is a central product and $C_{2^k}$ is a cyclic group of order $2^k$.
For example, I obtained that $Aut(D_{2^n}*C_{2^k})$ is a 2-group when $n\geq 4$ with a long calculation (if correct). But I think there should be a reference for that.
Any comment and well known fact about it is welcome.