  We prove foundational results for homological Dehn functions of groups of
type $FP_2$ such as superadditivity and the invariance under quasi-isometry. We
then study the homological Dehn functions of Leary's groups $G_L(S)$ providing
methods to obtain uncountably many groups with a given homological Dehn
function. This allows us to show that there exist groups of type $FP_2$ with
quartic homological Dehn function and unsolvable word problem.
