Is it true that if $G$ is a free group and $H$ is a subgroup of $G$ such that $[G:H]=n$ where $n>1$ then $[G^\text{ab}:H^\text{ab}]>1$ or can we have any property of $[G^\text{ab}:H^\text{ab}]$? Where $G^\text{ab}$ is the abelianization of $G$.
Edit (YCor Sep. 2023) as it has been pointed out, the question is not meaningful since $H^\text{ab}$ is not a subgroup of $G^\text{ab}$; more precisely, the canonical map $f:H^\text{ab}\to G^\text{ab}$ is not injective. However, the questions about the index of $f(H^\text{ab})$ in $G^\text{ab}$ make sense and have been addressed in the answers.