A polyhedron is snub quasi-regular if it is vertex-transitive with irregular
pentagonal, hexagonal, or octagonal vertex figures. Vertex transitivity means
that for any two vertices of the polyhedron, there exists a translation,
rotation, and/or reflection that leaves the outward appearance of the
polyhedron unchanged yet moves one vertex to the other. A vertex figure is
the polygon produced by connecting the midpoints of the edges meeting at the
vertex in the same order that the edges appear around the vertex. There are
only two snub quasi-regular polyhedra that are not self-intersecting, namely
the Snub Cube and the
Snub Dodecahedron.
When self-intersection is allowed, there are 10 other snub quasi-regular
polyhedra.

H. S. M. Coxeter and J. C. P. Miller discovered all 10 between 1930 and
1932 [1].
M. S. Longuet-Higgins and H. C. Longuet-Higgins rediscovered nine
(all except the Great Dirhombicosidodecahedron) between 1942 and 1944
[1].
J. Lesavre and R. Mercier described five (the Snub Dodecadodecahedron,
the Inverted Snub Dodecadodecahedron, the Great Snub Icosidodecahedron,
the Great Inverted Snub Icosidodecahedron, and the Great Retrosnub
Icosidodecahedron) in 1947 [2].

H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences246 (1954), 401-450.

[2]

J. Lesavre and R. Mercier, Dix Nouveaux Polyèdres Semi-régulièrs sans Plan de Symétrie, Comptes Rendus des Séances de l'Académie des Sciences224 (1947), 785-786.