Verb for process:$X\implies Y$ that only increases symmetries: $G(X) \subset G(Y)$.

Question

Let $X$, $Y$ be two objects, anything really. Suppose that some process, for instance a simple algorithm, converts $X$ into $Y$. Suppose also that both $X$ and $Y$ have associated symmetry groups. What is the verb describing this process if symmetries are (preserved, conserved, something else?): $G(X) \subset G(Y)$. In other words $Y$ is constructed by the process, from $X$, and no symmetries are lost. Please provide the best vocabulary term for describing that property. Thanks. $G(X) = $ the symmetries of $X$.