Let X = LP(Ii), where 1 ? p R,, is a uniformly bounded, strongly continuous representation of G in X by separation-preserving operators (for instance, by positivity-pre- serving operators). We show that R transfers to X strong type maximal estimates for sequences of convolution operators defined on LP(G). If R also has a uniformly bounded version in L2(%t), then R will transfer to LP((t) strong type bounds for maximal operators defined by sequences of multipliers which are continuous on the dual group G. These theo- rems, which cease to be valid if the separation-preserving hypothesis is removed, provide strong type maximal operator counterparts for the corresponding single operator theorems of Coifman-Weiss transference theory. One consequence of the second theorem described above is a maximal.theorem counterpart of the Homomorphism Theorem for Sin- gle Multiplier Transforms.