Characteristically DP-decomposable subgroup

Definition with symbols
A subgroup $$H$$ inside a group $$G$$ is said to be characteristically DP-decomposable if we can express $$G$$ as an internal direct product of subgroups $$G_w$$ such that $$H$$ is the internal direct product of all the $$H_w = H \cap G_w$$, and further, each $$H_w$$ is characteristic in $$G_w$$.

Stronger properties

 * Characteristic subgroup
 * Direct factor
 * CDF-subgroup

Weaker properties

 * Normal subgroup

Metaproperties
This actually follows from the way it's defined. The given property is the strongest direct product-closed subgroup property that is weaker than the property of being characteristic.

Clearly, the trivial subgroup, as well as the whole group, are characteristically DP-decomposable.