Generating pairs of projective special linear groups that fail to lift

The following problem was originally posed by B. H. Neumann and H. Neumann. Suppose that a group G can be generated by n elements and that H is a homomorphic image of G. Does there exist, for every generating n‐tuple (h1,…,hn) of H, a homomorphism ϑ:G→H and a generating n‐tuple (g1,…,gn) of G such that (g1ϑ,…,gnϑ)=(h1,…,hn)? M. J. Dunwoody gave a negative answer to this question, by means of a carefully engineered construction of an explicit pair of soluble groups. Via a new approach we produce, for n=2, infinitely many pairs of groups (G,H) that are negative examples to Neumanns' problem. These new examples are easily described: G is a free product of two suitable finite cyclic groups, such as C2*C3, and H is a suitable finite projective special linear group, such as PSL(2,p) for a prime p≥5. A small modification yields the first negative examples (G,H) with H infinite.