To tell/request/order to do (in such a way)
Structure
- Verb + ように + 言う / 頼む / 命じる
[Used to indirectly quote an order/request. 言う can be used for both, but 頼む is only for requests and 命じる for orders]
It says that this is used for indirect quoted, but in the example
僕(ぼく)のスマホを貸(か)してあげるけど、濡(ぬ)れないように頼(たの)むぞ。
it seems to me to be a direct request. Is it something to do with the construction of the larger sentence?
Probably a SUPER nitpicky thing to bring up, but I feel like either this grammar point should have 頼む & 命じる written in kana or the kanji/kana should be shown in the question since it’s more about teaching the grammar point and less about remembering vocab. Perhaps new users may get thrown off a bit? I only bring this up because I stupidly couldn’t remember 頼むtoday and wasn’t able to answer with the basic ように言う 
I think the example sentence
誰もママに何も言わないように頼んだがバレてしまった。
should have a comma after the が, so that in quizzes it’s clear that it isn’t meant to be the subject marking particle which would suggest nominalization is required. Compare
誰もママに何も言わないように頼んだことがバレてしまった。
with
誰もママに何も言わないように頼んだが、バレてしまった。