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?
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.
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