It will depend on the specific Charm/Immune to Charm effects
The problem here is that in the case of an ability that makes a creature unaffected by a certain condition, it says nothing about whether or not it's actively removing that condition.
There are specific ways to remove the Charmed condition whether it's Cleansing Touch, Dispel Magic, or other spells and abilities that do so.
This is where it gets a bit trickier. We can look at this question on a Paladin's Aura of Courage in relation to the frightened condition. The language of the Aura is similar to the language used in Infernal Command:
Aura of Courage
[creatures in the aura] can’t be frightened while you are conscious.
Infernal Command (Mordenkainen's Tome of Foes, p. 167)
Each ally of the narzugon within 60 feet of it can’t be charmed or frightened until the end of the narzugon’s next turn.
The answer provided in the linked question provides evidence that the Condition has not been removed, but it is only suppressed. Given the similarity in language with Infernal Command, the same answer should apply here as well.
Infernal Command will suppress the Charm/Frightened condition until the Narzugon's next turn.
Other abilities may work differently, but in the case of Infernal Command, this is likely the answer.
Ruling with Removing
Removing the condition entirely isn't necessarily a bad ruling. Crawford himself stated that the RAW was unclear, but went on to state the RAI. If your table prefers the complete removal the condition, the ambiguous RAW says that's okay - and it is. It just gives those suppressing options a bit more firepower, so folks should plan accordingly when designing encounters or acting within them.