Characteristicity satisfies partition difference condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., partition difference condition)
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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