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Is it possible to give a reasonable description of those finite solvable groups $G$ such that $A = {\rm Aut}(G)$ is also solvable? The central case to deal with is that in which $G$ is a $p$-group of ...

This is an enlarged version of my question on MSE. It was suggested I ask here instead. Suppose the finite group $G$ has exactly two conjugacy classes that are not self-inverse (a conjugacy class is ...

Edit: I'd be happy to hear any vague thoughts you might have, however far they may be from a complete solution! I asked this on math.stackexchange a couple of days ago, but it didn't attract any ...

During my researches, I've obtained a class of finite groups as follows. Let $\mathcal{C}$ be the class of all finite groups $G$ such that for every factorization $|G|=ab$ there exists a subgroup $H\...