Rank of sumsets in matroids

Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is G. Let $X$ and $Y$ be subsets of G, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H$=$X+Y$. We denote the rank function of $M$ by $r$. Then can we say that $r(X+Y)geq$ $r(X)+r(Y)-r(H)$?
(A similar result holds for the cardinality of sumsets of a given group, well-known as Kneser’s theorem)