The most common and most likely answer to this crossword clue is the 9 letter word EXECUTIVE
I've seen this crossword clue in The Daily Telegraph
E
X
E
C
U
T
I
V
E
More clues leading to the result EXECUTIVE
- Director put to death with intravenous injection
- Around 4, take out the person responsible
- Person responsible put to death about four
- Management finish around four
- Old style I incorporated in evening suit
- Former English beauty accommodating five administrators
- Government kill with intravenous injection
- Kill around four of those in management?
- Administrator to perform around the four with...
- English team has English quota and very 'English' boss
Buddy explains!

The straight part
We know that "those in management" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case those in management = executive.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 EXECUTIVE mean?
EXECUTIVE noun- A chief officer or administrator, especially one who can make significant decisions on their own authority.
- The branch of government that is responsible for enforcing laws and judicial decisions, and for the day-to-day administration of the state.
- A process that coordinates and governs the action of other processes or threads; supervisor.
- Designed or fitted for execution, or carrying into effect.
- Of, pertaining to, or having responsibility for the day-to-day running of an organisation, business, country, etc.
- Exclusive.