  To a set $\mathcal{B}$ of 4-subsets of a set $\Omega$ of size $n$ we
introduce an invariant called the `hole stabilizer' which generalises a
construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based
on Loyd's `15-puzzle'. It is shown that hole stabilizers may be regarded as
objects inside an objective partial group (in the sense of Chermak). We
classify pairs $(\Omega,\mathcal{B})$ with a trivial hole stabilizer, and
determine all hole stabilizers associated to $2$-$(n,4,\lambda)$ designs with
$\lambda \leq 2$.
