The most common and most likely answer to this crossword clue is the 7 letter word ANTLERS
I've seen this crossword clue in The Daily Telegraph
A
N
T
L
E
R
S
Buddy explains!
The straight part
We know that "Those beastly overheads" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case those beastly overheads = antlers.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 ANTLERS mean?
ANTLERS noun- A branching and bony structure on the head of deer, moose and elk, normally in pairs. They are grown and shed each year. (Compare with horn, which is generally not shed.)