  In this paper, we investigate the moduli of surfaces of general type
admitting genus 2 fibrations with irregularity q = g_b + 1, where g_b >= 2 is
the genus of the base. We prove that smooth fibrations are parametrized by a
unique component in the moduli space. The same result applies to nonsmooth
fibrations with special values of g_b. In the general case, we give a bound on
the dimension of the corresponding connected components.
