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Jesper Grodal and I once looked at this, cf. this answer. 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 extension 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.