  In this paper we classify all finite 2-groups of class 2 for which every
automorphism of order 2 leaving the Frattini subgroup elementwise fixed is
inner. We prove that every such group G is isomorphic to Q(n; r) = <a, b|
a^{2n}= b^{2r}= 1; a^2^{n-r}= [a, b]> for some positive integers r; n such that
2 < 2r <= n; and every automorphism of Q(n; r) of order 2 leaving the Frattini
subgroup elementwise fixed is inner.
