Automorphisms of framed operads

Let $\mathsf{P}$ be an operad acted upon by a group $G$, and let $\mathsf{Q}=\mathsf{P}⋊G$ be the corresponding framed operad. We relate the homotopy automorphism groups of $\mathsf{P}$ and $\mathsf{Q}$. We apply the result to compute the automorphisms of the framed little disks operad.