Impossible Functors, Impossible Natural Transformations

It turns out that there are only trivial functors between many commonly occurring categories. For instance, the only functors from the category of groups to the category of finite groups, or from the category of countable groups to the category of finitely generated groups, are the constant functors. We explore some general statements along these lines. 

Moreover, the image of augmented functors FX→X or co-augmented endo-functors X→FX, say,  in the category of groups, is highly constrained. For example, while there is a natural abelian quotient of a group, there is no natural abelian subgroup and there is no natural perfect group quotient.  We examine strong restrictions on the possible image of such (co-)augmented functors.

This leads to the question: what are the infinity categorical extensions of these rather transparent statements? Can one naturally map an infinite loop space to a pointed space?