Joel Hamkins has left four open questions about automorphism tower theorem in his wonderful paper Every group has a terminating transfinite automorphism tower.
In fact, the four questions are "For which ordinals $\gamma$ is there a group whose tower becomes centerless in exactly -a many steps? Is there a countable group with an uncountable automorphism tower? Is there a finite group with an uncountable automorphism tower? Is there a finite group G such that $G_\omega$ is infinite?"
I'd like to ask that since the paper has been written (which was published in 1999), has there been progress made in answering these four or relative questions.
Hamkins, Joel David, Every group has a terminating transfinite automorphism tower, Proc. Am. Math. Soc. 126, No. 11, 3223-3226 (1998). ZBL0904.20027.
