Automorphism groups of tetravalent Cayley graphs on minimal non-abelian groups

A Cayley graph $X=\mathrm{Cay}(G,S)$ is called normal for $G$ if the right regular representation $R(G)$ of $G$ is normal in the full automorphism group $\mathrm{Aut}(X)$ of $X$. In the present paper it is proved that all connected tetravalent Cayley graphs on a minimal non-abelian group $G$ are normal when $(|G|, 2)=(|G|,3)=1$, and $X$ is not isomorphic to either Cay$(G,S)$, where $|G|=5^n$, and $|\mathrm{Aut}(X)|=2^m.3.5^n$, where $m \in \{2,3\}$ and $n\geq 3$, or Cay$(G,S)$ where $|G|=5q^n$ ($q$ is prime) and $|\mathrm{Aut}(X)|=2^m.3.5.q^n$, where $q\geq 7$, $m \in \{2,3\}$ and $n\geq 1$.