  Let G be a finite group. An element x in G is a real element if x is
conjugate to its inverse in G. For x in G, the conjugacy class x^G is said to
be a real conjugacy class if every element of x^G is real. We show that if 4
divides no real conjugacy class size of a finite group G, then G is solvable.
We also study the structure of such groups in detail. This generalizes several
results in the literature.
