The cost of symmetry for tailed stars

It is known that if $n$ vertices can be removed from a connected graph $\mathrm{\Gamma }$ so that no subgraphs isomorphic to the graph $K$ remain, then no more than $\mathrm{\mid }V(K)\mathrm{\mid }\cdot n$ vertices can be removed, forming a set invariant with respect to all automorphisms of the graph $\mathrm{\Gamma }$, so that no subgraphs isomorphic to the graph $K$ remain. We construct an infinite set of (connected) graphs $K$ for which this estimate is not exact.