Characteristicity satisfies partition difference condition

Statement
Suppose $$G$$ is a group and $$H$$ is a subgroup of $$G$$ such that $$H$$ has a partition into subgroups $$H_i, i \in I$$, with $$I$$ containing at least two elements. Then, if all except possibly one of the $$H_i$$ is a characteristic subgroup of $$G$$, all the $$H_i$$s are characteristic subgroups of $$G$$.

Related facts

 * Union of two subgroups is not a subgroup unless they are comparable
 * Normality satisfies partition difference condition
 * Full invariance does not satisfy partition difference condition