Verbally closed subgroup

Definition
Suppose $$G$$ is a group and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is verbally closed in $$G$$ if the following is true: For any word $$w$$ in $$n$$ letters, the image of $$H^n$$ under the word map corresponding to $$w$$ equals the intersection of $$H$$ with the image of $$G^n$$ under the word map corresponding to $$w$$.