On groups automorphisms fixing subnormal subgroups setwise

We study the groups $\Aut_{sn}(G)$, $\Aut_d(G)$, $\Aut_ \chi(G)$ of all automorphisms of a group $G$ fixing all subgroups of $G$ with the property of being: subnormal, subnormal of defect at most $d$, or lying between a characteristic subgroup of $G$ and its derived subgroup, respectively. We give some sufficient conditions for such groups to be parasoluble with length at most $2$ or $3$.