Quotient-torsion-freeness-closed subgroup

Definition
Suppose $$G$$ is a group and $$H$$ is a normal subgroup of $$G$$. We say that $$H$$ is a quotient-torsion-freeness-closed subgroup of $$G$$ if $$H$$ is a normal subgroup of $$G$$ and for any prime number $$p$$ such that $$G$$ is $$p$$-torsion-free, the quotient group $$G/H$$ is also $$p$$-torsion-free.