We consider isomorphism "up to a change of language," which we call twisted isomorphism. We call a structure twistable
if it has a twisted automorphism which is not an automorphism.
We consider three questions: \( (a) \) in what contexts can we classify the twistable structures; \( (b) \) when do their twisted automorphism groups split
over the ordinary automorphism group; \( (c) \) when does the action of the twisted automorphism group on the automorphism group induce the full
automorphism group of the latter? The last two questions are suggested by work of Cameron and Tarzi.
My student Rebecca Coulson has classified the twistable metrically homogeneous graphs. I will explain what these are, and what is known about these problems in that setting.