$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $\Sp(4,\mathbb{Z})$ that are conjugate in $\GL(4,\mathbb{Q})$ $\big($or in $\SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it follow that their indices are equal: $[\Sp(4,\mathbb{Z}):G_1]=[\Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!