beckon to, command
麾 centers on directing others by gesture or authority: beckoning someone to come, or giving a command.
麾 combines 麻 (hemp) and 毛 (hair/feather), likely referring to a feathered or tasseled banner used to signal commands. The exact historical development is uncertain, but the character is associated with directing or beckoning.
Under a hemp canopy 麻, a feathered banner 毛 is raised to beckon troops and issue commands.
For キ, imagine a commander beckoning with a key-shaped banner: key -> キ, and the key gesture commands attention.
under the command (of); under the banner (of); influenced (by)