Jesper Grodal and I once looked at this, cf. [this answer][1]. In particular Eilenberg and MacLane constructs a universal obstruction in $H^3(\text{Out}(G);Z(G))$ for the extension $1\rightarrow G\rightarrow \widetilde{\text{Aut}}(G) \rightarrow \text{Out}(G) \rightarrow 1$ to exists. A good reference is Brown's "Cohomology of groups". For small groups one can compute this obstruction explicitly, it turns out to be non-zero for the group $D_{16}$. As this has center $C_2$ question number 3 unfortunately also has a negative solution.


  [1]: https://mathoverflow.net/questions/81721/a-homotopy-commutative-diagram-that-cannot-be-strictified/82958#82958