The most common and most likely answer to this crossword clue is the 7 letter word INQUIRE
I've seen this crossword clue in The Guardian
I
N
Q
U
I
R
E
Buddy explains!

The straight part
We know that "Make investigation" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case make investigation = inquire.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does INQUIRE mean?
INQUIRE verb- To ask (about something).
- To make an inquiry or an investigation.
- To call; to name.