Here's an elementary observation, in the definition of non-splitting you can restrict your attention to those H's generated by a two (not necessarily distinct) elements of c.
In particular, if no powers of elements of c are in c (for example, if c consists of involutions) and any two distinct elements of c generate all of G then it follows that (G,c) is nonsplitting.
As FC points out two involutions always generate a dihedral group. As JSE points out (G,c) for c a class of involutions is splitting nonsplitting iff the resulting dihedral groups are all of 2*odd order. In particular, (G,c) is splitting nonsplitting for c an involution iff the product of any two elements of c has odd order.
