Half guard is quite a variable position. There's a classic half guard system that I call 1-2-3, 1-2-3.
1) Secure leg ride/lock down.
2) Get under hooks, or at least an underhook on the same side as the leg ride.
3) Shrimp or leg lift onto your side.
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1) Threaten the back take.
2) If they whizzer hard, roll them over (Plan B).
3) If they keep their weight back, get to your knees and wrestle them over (old school).
I just pulled this series off at the London Open at the weekend in the absolute division against a guy who won gold 2 weight classes up and won the match.
There are lots of different types of half guard like deep half, quarter guard, underhooking the opposite leg for electric chair, kneebar etc. I really believe in systems rather than single techniques.