I've seen this kind of supervision happen in three ways, the first one was confirmed by an ILE:
- EIIs are not very emotive and outwardly expressive, which makes it hard for an ILE to read them, understand what they are feeling and their relationship to the ILE. This has that on-going background supervision effect that the EIIs don't seem to notice: it makes the ILE feel particularly self-conscious around the EII. They can't tell if you're mad at them, if you're sad or happy, do you love them or hate them, and this is unsettling for a Fe HA type. They may try to entertain the EII to make the EII "lighten up" as they sometimes misinterpret the Fi seriousness as the person being angry with them or having a bad day, only to be perplexed that this doesn't elicit the kind of reaction that they need.
- ILEs will in some shape or form test people's wits and gauge their intelligence levels, towards which end they may start debates but also sometimes they intentionally mislead others, trying to guide their thoughts a certain way, and play the trickster part. EIIs clamp down on this, as they may find incessant verbal arguing to be annoying and upsetting, and the attempts to trick and mislead others as unethical.
- EIIs especially of Fi-subtype are much more precise and concrete in their reasoning. They don't let ILEs get away with vague statements and may nitpick on something they've said, complain about the vagueness, and generally seem to hold them to a higher standard in this regard (this is more applicable for intuitive ILE subtype).
I don't think supervision is really all that one-sided as the classical descriptions make them sound. The EII cannot keep up with the ILE in terms of extraversion and their creative function - when it comes to orienting in new situations and handling new information the ILE does this quicker with greater flexibility and logical discernment than the EII - so it's not like this relationship turns into one that resembles the literal meaning of its name. The supervisee usually has at least those two points where they are able to outperform their supervisor and make a positive impression on them.