I was wondering if it is possible to determine all $n$ such that there exists a group $G$ with odd $|\operatorname{Aut}(G)|>1$. The following numbers are some examples that occur in the list:
$729=3^6$ (by direct computation);
$p^5$ for odd primes $p$ (from Finite groups with odd order automorphism groups);
$3p^5$ and $3p^6$ for primes $p\equiv 1\pmod 3$ (from Complete groups of order $3p^6$),
... and so on.
If the concrete list cannot be given, can we say something about the prime factors? For example, given a set of odd primes $\{p_1,\cdots,p_r\}$, does there exists a group $G$ such that the prime factors of $|G|$ are exactly those $p_1,\cdots,p_r$ and that $|\operatorname{Aut}(G)|$ is odd?
Any help/reference appreciated.
Edit: Those even orders are twice the odd orders. If $|G|$ is even and $|\operatorname{Aut}(G)|$ is odd, then $G=G_0\times C_2$ with $|G_0|$ odd and $|\operatorname{Aut}(G_0)|$ odd, as stated in this answer. Thanks Dietrich Burde for providing this link in the comments.