**Abstract**

In Part I (arXiv:1805.03917) we showed that the moduli spaces of n mutually isotopic, embedded copies of a closed manifold P in a connected, open manifold M parametrised modulo an open subgroup G of the diffeomorphism group of P are homologically stable as n goes to infinity, as long as a certain condition on the relative dimensions of M and P is satisfied.

In Part II (this paper) we use this to deduce homological stability results for symmetric diffeomorphism groups of manifolds, with respect to the operation of *parametrised connected sum* (for example, surgery). As a special case of this, we see that the diffeomorphism groups of manifolds with certain conical singularities are homologically stable with respect to the number of singularities of a fixed type.

To do this, we first extend the main result of Part I to moduli spaces of disconnected, *labelled* submanifolds — in which the components of the submanifolds are equipped with labels in a fibration over an embedding space. This in turn is a corollary of *twisted* homological stability for moduli spaces of disconnected, unlabelled submanifolds with finite-degree coefficients, which we prove using the main (untwisted) result of Part I as an input.