This question is essentially a reposting of this question from Math.SE, which has a partial answer. YCor suggested I repost it here.

Our starting point is a theorem of Matumoto: every group $Q$ is the outer automorphism group of some group $G_Q$ [1]. It seems to be a research theme to place restrictions on the groups involved. For example, Bumagin and Wise proved that if we restrict $Q$ to be countable then we may take $G_Q$ to be finitely generated [2], and more recently Logan proved that if we restrict $Q$ to be finitely generated and residually finite group then we may take $G_Q$ to be residually finite [3, Corollary D] (this paper also cites quite a few other papers which play this game).

However, all the results I have found always produce infinite groups $G_Q$, even when the "input" groups $Q$ are finite. For example, Matumoto's groups $G_Q$ are fundamental groups of graphs of groups (so are always infinite), Bumagin and Wise use a variant of Rips' construction (so (as $Q$ is finite) their groups $G_Q$ have finite index in metric small cancellation groups, so are infinite), and Logan's groups $G_Q$ are HNN-extensions of hyperbolic triangle groups (so again are infinite). So we have a question:

Does every finite group $Q$ occur as the outer automorphism group of some finite group $G_Q$?

The answer is "yes" if we take $Q$ to be finite abelian or a symmetric group; this is what the answer to the original Math.SE question proves.

[1] Matumoto, Takao. "Any group is represented by an outerautomorphism group." Hiroshima Mathematical Journal 19.1 (1989): 209-219. (Project Euclid)

[2] Bumagin, Inna, and Daniel T. Wise. "Every group is an outer automorphism group of a finitely generated group." Journal of Pure and Applied Algebra 200.1-2 (2005): 137-147. (doi)

[3] Logan, Alan D. "Every group is the outer automorphism group of an HNN-extension of a fixed triangle group." Advances in Mathematics 353 (2019): 116-152. (doi, arXiv)