It looks to me as if this will not work with $3$-cycles in $G = A_{n}.$ Suppose, for example, that $n =2m$ and take $H=A_{m}.$ Then $G$ contains $\frac{n(n-1)(n-2)}{6}$ $3$-cycles, and $H$ contains $\frac{m(m-1)(m-2)}{6}$ $3$-cycles, so about $\frac{1}{8}$ of the $3$-cycles in $G.$ However, $[G:H] \to \infty$ as $m \to \infty.$

(Leter edit in response to question in commentI think it also fails for transvections in $G = {\rm GL}(2m,2)$ and $H = {\rm GL}(2m-1,2).$ The number of transvections in $G$ is ( I think) $2^{4m-1} - 2^{2m-1}$ and the number of transvections in $H$ is $2^{4m-3} - 2^{2m-2}.$ For large $m,$ the proportion of tranvections from $G$ which lie in $H$ is around $\frac{1}{4}$ but $[G:H] \to \infty$ as $m \to \infty$.

It looks to me as if this will not work with $3$-cycles in $G = A_{n}.$ Suppose, for example, that $n =2m$ and take $H=A_{m}.$ Then $G$ contains $\frac{n(n-1)(n-2)}{6}$ $3$-cycles, and $H$ contains $\frac{m(m-1)(m-2)}{6}$ $3$-cycles, so about $\frac{1}{8}$ of the $3$-cycles in $G.$ However, $[G:H] \to \infty$ as $m \to \infty.$

(Later edit in response to question in comment. I think it also fails for transvections in $G = {\rm GL}(2m,2)$ and $H = {\rm GL}(2m-1,2).$ The number of transvections in $G$ is ( I think) $2^{4m-1} - 2^{2m-1}$ and the number of transvections in $H$ is $2^{4m-3} - 2^{2m-2}.$ For large $m,$ the proportion of tranvections from $G$ which lie in $H$ is around $\frac{1}{4}$ but $[G:H] \to \infty$ as $m \to \infty$.

It looks to me as if this will not work with $3$-cycles in $G = A_{n}.$ Suppose, for example, that $n =2m$ and take $H=A_{m}.$ Then $G$ contains $\frac{n(n-1)(n-2)}{6}$ $3$-cycles, and $H$ contains $\frac{m(m-1)(m-2)}{6}$ $3$-cycles, so about $\frac{1}{8}$ of the $3$-cycles in $G.$ However, $[G:H] \to \infty$ as $m \to \infty.$

It looks to me as if this will not work with $3$-cycles in $G = A_{n}.$ Suppose, for example, that $n =2m$ and take $H=A_{m}.$ Then $G$ contains $\frac{n(n-1)(n-2)}{6}$ $3$-cycles, and $H$ contains $\frac{m(m-1)(m-2)}{6}$ $3$-cycles, so about $\frac{1}{8}$ of the $3$-cycles in $G.$ However, $[G:H] \to \infty$ as $m \to \infty.$

(Leter edit in response to question in commentI think it also fails for transvections in $G = {\rm GL}(2m,2)$ and $H = {\rm GL}(2m-1,2).$ The number of transvections in $G$ is ( I think) $2^{4m-1} - 2^{2m-1}$ and the number of transvections in $H$ is $2^{4m-3} - 2^{2m-2}.$ For large $m,$ the proportion of tranvections from $G$ which lie in $H$ is around $\frac{1}{4}$ but $[G:H] \to \infty$ as $m \to \infty$.