The Sylow Theorems · Hard problem

$\operatorname{Aut}(\mathbb Z_{11}) \cong U(11)$ is cyclic of order $10$. A nontrivial order-$5$ automorphism $a \mapsto a^k$ (used to build the non-abelian group of order $55=5\cdot11$) exists for exactly $4$ residues $k$ with $2\le k\le10$. Find the SUM of all $4$ such $k$.