  The study of curvature properties of homogeneous Finsler spaces with
$(\alpha, \beta)$-metrics is one of the central problems in Riemann-Finsler
geometry. In the present paper, the existence of invariant vector fields on
homogeneous Finsler spaces with square $(\alpha, \beta)$-metric and Randers
changed square $(\alpha, \beta)$-metric is proved. Further, an explicit formula
for $S$-curvature of these $(\alpha, \beta)$-metrics is established. Finally,
using the formula of $S$-curvature, the mean Berwald curvature of afore said
$(\alpha, \beta)$-metrics is calculated.
