The most common and most likely answer to this crossword clue is the 5 letter word LYNCH
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Buddy explains!
The straight part
We know that "undertake execution without trial" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case undertake execution without trial = lynch.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 LYNCH mean?
LYNCH verb- To execute (somebody) without a proper legal trial or procedure, especially by hanging and backed by a mob.
- A ledge, a terrace; a right-angled projection; a lynchet.
- An acclivity; a small hill or hillock.