Generally, for every crime x, the act of planning x is a crime known as "conspiracy to commit x"

Well, they are accused of not investigating possible espionage in their own army. Lacking details, it is really hard to say what is the exact situation, but obviously they are not interested in preventing crimes.

The community here is understandably very anti-surveillance, but you're right about the purpose of intelligence.

When we talk about intelligence and counter-terrorism, it's more about preventing terrorist attacks than reacting to them.

The UK is more open with statistics than the US — their most senior counter-terrorism officer claims that they foiled 22 terrorist attacks from March 2017 to September 2019 [1].

[1] https://www.theguardian.com/uk-news/2019/sep/09/foiled-terro...

This would generally be the function of a counter-intelligence or internal (criminal) investigation group.

In the US that would be handled by the FBI -- they're legally the only ones that can spy on citizens in in the US. They work hand-in-glove with other agencies, but they own domestic.

A non-US example would be Russia, where the FSB (which Putin used to be the head of) handles internal investigations and counterintelligence like the FBI, while the SVR handles foreign intelligence. Or MI5 vs MI6 in the UK, etc.