If I said “he did X to me,” I would probably say 彼にXされた. On the other hand, if
I said “I let him do X,” I would say 彼にXさせた. But if I were to combine the two
and say 彼にXさせられた, that would mean “he made me do X,” correct? How then would I
be able to say “I let him do X to me”?

EDIT: for a more concrete example, say I was in a class and we were made to
form groups. Either we chose our groupmates ourselves, or we let the teacher
group us. Say I let the teacher put me in a group: how would I say “I let the
teacher put me in a group?” (I imagine saying “the teacher put me in a group”
would involve passive voice, me being the narrator and therefore the subject
of the sentence).
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passive-voice causation permission
How to say “I let him do X to me”?
You can't really use させられる I think because this form has a special meaning. It
implies that someone (other than you or your close circle) forced you to do
something you don't want to do, with a nuance that you are the victim of the
action (which is negative of course since you are a victim). It can not be
used to express "to let someone do X to me".

The most suitable structure would be a casual causative form. So your example
would give :

  * 先生は(私に)グループに入らせた

which translated would be something like :

  * I let the teacher put me into a group
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I think you could express this idea by saying that you,

  * " **had your teacher choose** the group you would be in **for you** "

> 私が入るグループを **先生に選んでいただいた** 。
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In the specific example you give, I'd probably just say 「私のグループは先生に任せた」 ("I
left my group up to the teacher") or similarly 「グループの選択は先生に任せた」("I left the
choice of my group to the teacher").

In the case you provide, it's more like you're saying you're leaving something
up to someone else, rather than allowing someone to do something _to you._

If you're looking for something more general than "leaving it up to someone
else," you'd probably want something of the form ～してもいい (it's ok even if you
do X) as in:

```

    彼なら私に何をてもいいのよ。
    
```

("It's OK no matter what he does to me.")

Or:

```

    「私を触ってもいい。」
    
```

("It's fine if [it] touches me.")

If you want a more active voice, you could use ～頼む, as in:

```

    「グループを選ぶのを先生に頼んだ」
    
```

("I asked my teacher to pick my group")

A more direct translation might be 「～こと許す」("I will allow X [to be done]"):

```

    私の頭をなでること許す。
    
```

("I will let you pat my head.")

But, at least to me, this seems to have a bit of a "high and mighty" feel to
it, as though you're looking down on the person and doing them a favor by
allowing the action, and it has a very active voice.
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I think this fits in your question,

> 先生に僕が入るグループを **選んでもらった**

  * I **let/would like/wished/hoped** teacher choose to be in a group.

If you put stress **もらった** , it gets close the nuance of "let" in English.
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I assume the question used the versatile word "let" to mean the teacher
becomes the decision maker by choice, and _not_ by command, which is otherwise
rude.

Personally, I would avoid the direct translation. The following suggested
expressions use `私のグループは` that refers to the group I would be in (translated
as "my grouping"); not the literal "my group", which is yet to exist in this
context.

> A. 私のグループは 先生に決めてもらった｡  
>  A. [I] had [my] teacher decide for my grouping.
>
> B. 私のグループは 先生に決めてもらう方にした｡  
>  B. [I] opt for the choice that [my] teacher would decide for my grouping.

`A` is the straightforward, passive voice. The phrase `先生にきめてもらった` is probably
the shortest phrase one can use readily to describe that "the teacher decide
(do) the group (X) for me".

`B` is the alternative expression made by modifying `A`, which turns the
passive voice to not-so-active voice. As a result, `B` would be perceived less
straightforward, but would appeal the speaker as slightly independent (on the
surface only, because the teacher is still the decision maker by choice).

Note that `B` will make sense only when "the teacher would decide my grouping"
is available as a choice, due to the phrase `〇〇方にした` being used to refer "that
choice". In contrast, `A` will make sense regardless of that is a choice or
not.

Then again, these expressions may vary when describing the "do": If the groups
were already formed, the teacher would choose (選ぶ) which group the student (I)
would be in; else if no groups were formed yet, the teacher would decide (決める)
how the groups would be formed, or decide which and which students would be
put into the groups.
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