The smallest example is 2×AΓL(1,8) of order 336. A fairly standard permutation representation is generated by (+,-), (∞,0)(1,3)(2,6)(4,5), (1,2,3,4,5,6,7), and (1,2,4)(3,6,5), or you can use more GAP friendly points to get (9,10), (8,7)(1,3)(2,6)(4,5), (1,2,3,4,5,6,7), and (1,2,4)(3,6,5). AΓL(1,8) is a complete, solvable group with no normal subgroup of index 2.
I just checked the groups ordered by size, skipping nilpotent groups since they always have outer automorphisms. Probably one could be a bit smarter, looking for complete groups with no 2-quotients, but I wasn't sure the smallest example would be of the form 2×Complete.
There are examples like 2.Sz(8) which have no outer automorphisms, have a non-trivial center, but where the quotient by the center is not complete (so it is definitely not 2×Complete).