Isometries of $p$-convexified combinatorial Banach spaces

We show that if $1<p\neq 2<\infty$, then any isometry of the $p$-convexification of the combinatorial Banach space associated with a hereditary family of finite subsets of $\mathbb{N}$ containing the singletons is given by a signed permutation of the canonical basis. In the case of a generalized Schreier family, the result also holds for $p=2$, and every isometry is diagonal. These results are deduced from more general theorems concerning combinatorial-like Banach spaces.