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A mildly useful keyword might be Zappa-Szép product -- this is the case where $A$ and $B$ are both taken to be subgroups. See Wikipedia: en.wikipedia.org/wiki/Zappa–Szép_product (MathOverflow breaks the link at the -, so copy and paste into your browser unless you're interested in Frank, rather than Guido, Zappa.)
If $S_V$ acts transitively on $\Delta$, then I suppose $\Delta$ is some skeleton of a simplex, hence shellable. But that's almost surely not what you have in mind.
Indeed, one could view algebraic shifting as saying that any simplicial complex can be slightly perturbed (for a particular algebraic, complicated notion of 'perturbed') to become a bouquet of spheres.
The number of ways of decomposing an n-cycle into n-1 transpositions is the same as the number of spanning trees of a complete graph on n vertices. Indeed, given a such a decomposition of an n-cycle, one can build a spanning tree by putting an edge {i,j} iff the transposition (i,j) is present. (Verifying that this correspondence is invertible seems to take a little more work.)
Indeed, the positive answer is strongly hinted at by the fact that degenerate mappings are allowed in the definition of singular homology. (Hatcher even says that the name "singular" came about because singularities are allowed in the mapping.)
From what I can see, Kozlov defines an acyclic matching on a poset (which he calls a discrete Morse matching by analogy with later interest) and gives some structure theorems about such matchings. You don't get any of the nice homotopy theory consequences -- indeed, he seems to save any discussion of consequences for 11.2, where he links with topology of CW-complexes. So yes, technically, he calls something discrete Morse theory for posets; but it's interesting mainly/only for face lattices.
