TITLE: Non-split Aut(G) $\to$ Out(G)?
QUESTION [14 upvotes]: I'm looking for examples of outer automorphisms of a finite group $G$ which do not lift to automorphisms (i.e. non-split quotient map $\mathrm{Aut}(G)\to \mathrm{Out}(G)$, where $\mathrm{Out}(G) = \mathrm{Aut}(G)/\mathrm{Inn}(G)).$
I'm aware of the well-known $A_6$ (also $S_6$?) example, as explained in this Wikipedia article.  The article also mentions $PSL(2.q^2)$ for $q$ odd.
Are these the smallest examples?
Is there a complete (or at least longer) list of examples somewhere?

REPLY [6 votes]: A variation on Geoff's comment: Griess has proved that it's usually nonsplit for extraspecial $2$-groups:

Robert L. Griess, Jr., Automorphisms of extra special groups and
  nonvanishing degree 2 cohomology, Pacific Journal of
  Mathematics Vol. 48, No. 2 (1973) pp 403-422 doi:10.2140/pjm.1973.48.403,
  (pdf)