The linear programming relaxation permutation symmetry group of an orthogonal array defining integer linear program

There is always a natural embedding of $S_s\wr S_k$ into the linear programming (LP) relaxation permutation symmetry group of an orthogonal array integer linear programming (ILP) formulation with equality constraints. The point of this paper is to prove that in the $2$ level, strength $1$ case the LP relaxation permutation symmetry group of this formulation is isomorphic to $S_2\wr S_k$ for all $k$, and in the $2$ level, strength $2$ case it is isomorphic to $S_2^k\rtimes S_{k+1}$ for $k\geq 4$. The strength $2$ result reveals previously unknown permutation symmetries that can not be captured by the natural embedding of $S_2\wr S_k$. We also conjecture a complete characterization of the LP relaxation permutation symmetry group of the ILP formulation.