Almost simple groups with socle Ln(q) acting on Steiner quadruple systems

Let N=Ln(q), n⩾2, q a prime power, be a projective linear simple group. We classify all Steiner quadruple systems admitting a group G with N⩽G⩽Aut(N). In particular, we show that G cannot act as a group of automorphisms on any Steiner quadruple system for n>2.