# Characterising extendable automorphisms

Suppose H is a subgroup of a finite group G. Can the group of all automorphisms of H that extend to G can be characterized somehow? What condition on the group extension would guarantee that any automorphism of H can be extended to G?

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You might also be interested in: mathoverflow.net/questions/7782/… –  Sam Nead Dec 25 '09 at 21:07

An abstract answer to the question for all groups is given in the papers below. I have not followed the field in recent years. There may be other papers specific to finite groups.

Charles Wells, Automorphisms of Group Extensions, 1970.

Kung Wei Yang Isomorphisms of group extensions. Pacific J. Math. Volume 50, Number 1 (1974), 299-304.

D.J.S. Robinson, Applications of cohomology to the theory of groups, Groups – St. Andrews 1981, London Math. Soc. Lecture Notes vol. 71 (1982), pp. 46–80.

Jin Ping, Automorphisms of groups Journal of Algebra Volume 312, Issue 2, 15 June 2007, Pages 562-569

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thank you very much! –  Sebastian Burciu Dec 26 '09 at 8:58
Can we get a summary of the characterization put up here? –  Joel David Hamkins Dec 26 '09 at 14:25