Let G be a group and X be a nonempty set and H be a subgroup of G. For a given ${phi}_G;:;G{ imes}X{ ightarrow}X$, a group action of G on X, we define ${phi}_H;:;H{ imes}X{ ightarrow}X$, a subgroup action of H on X, by ${phi}_H(h,x)={phi}_G(h,x)$ for all $(h,x){in}H{ imes}X$. In this paper, by considering a subgroup action of H on X, we have some results as follows: (1) If H,K are two normal subgroups of G such that $H{subseteq}K{subseteq}G$, then for any $x{in}X$ ($orb_{{phi}_G}(x);:;orb_{{phi}_H}(x)$) = ($orb_{{phi}_G}(x);:;orb_{{phi}_K}(x)$) = ($orb_{{phi}_K}(x);:;orb_{{phi}_H}(x)$); additionally, in case of $K{cap}stab_{{phi}_G}(x)$ = {1}, if ($orb_{{phi}_G}(x);:;orb_{{phi}H}(x)$) and ($orb_{{phi}_K}(x);:;orb_{{phi}_H}(x)$) are both finite, then ($orb_{{phi}_G}(x);:;orb_{{phi}_H}(x)$) is finite; (2) If H is a cyclic subgroup of G and $stab_{{phi}_H}(x){ eq}$ {1} for some $x{in}X$, then $orb_{{phi}_H}(x)$ is finite.