Here's a case where $G$ and $H$ can be conjugate. First some notation: given a sequence ${k_n}$ \{k_n\}$of positive integers, let$[k_1,k_2,\ldots]$denote the permutation $$(1,\ldots,k_1)(k_1+1,\ldots,k_1+k_2)(k_1+k_2+1,\ldots,k_1+k_2+k_3)\cdots$$ with cycles of size$k_1,k_2,k_3\ldots$. For example,$[1,1,1,1,\ldots]$denotes the identity,$[2,2,2,2,\ldots]$denotes$(1,2)(3,4)(5,6)(7,8)\cdots$, and$[2,3,2,3\ldots]$denotes$(1,2)(3,4,5)(6,7)(8,9,10)\cdots$. Let $$g = [1,2,\;\;1,2,4,\;\;1,2,4,8,\;\;\ldots],$$ let $$h = [1,1,1,\;\;1,1,1,2,2,\;\;1,1,1,2,2,4,4,\;\;\ldots],$$ and let$G$and$H$be the cyclic subgroups generated by these elements. Since$g$and$h$have the same cycle structure, they are conjuagte in$Sym(\mathbb{N})$, so$G$and$H$are conjugate subgroups. However, for sufficiently large$n$, the orbit of$(\pi(1),\pi(2),\ldots,\pi(n))$under$G$will be precisely twice the size of the orbit under$H$. Of course, in this example$G$and$H$both have infinitely many orbits of size$2^k$for every$k$, so this does not answer the more restrictive version of the question. 1 Here's a case where$G$and$H$can be conjugate. First some notation: given a sequence${k_n}$of positive integers, let$[k_1,k_2,\ldots]$denote the permutation $$(1,\ldots,k_1)(k_1+1,\ldots,k_1+k_2)(k_1+k_2+1,\ldots,k_1+k_2+k_3)\cdots$$ with cycles of size$k_1,k_2,k_3\ldots$. For example,$[1,1,1,1,\ldots]$denotes the identity,$[2,2,2,2,\ldots]$denotes$(1,2)(3,4)(5,6)(7,8)\cdots$, and$[2,3,2,3\ldots]$denotes$(1,2)(3,4,5)(6,7)(8,9,10)\cdots$. Let $$g = [1,2,\;\;1,2,4,\;\;1,2,4,8,\;\;\ldots],$$ let $$h = [1,1,1,\;\;1,1,1,2,2,\;\;1,1,1,2,2,4,4,\;\;\ldots],$$ and let$G$and$H$be the cyclic subgroups generated by these elements. Since$g$and$h$have the same cycle structure, they are conjuagte in$Sym(\mathbb{N})$, so$G$and$H$are conjugate subgroups. However, for sufficiently large$n$, the orbit of$(\pi(1),\pi(2),\ldots,\pi(n))$under$G$will be precisely twice the size of the orbit under$H$. Of course, in this example$G$and$H$both have infinitely many orbits of size$2^k$for every$k\$, so this does not answer the more restrictive version of the question.