Different inversions “require” different body musculo-skeletal synergies. Since you don’t mention the particular inversions of interest I’m short-sheeted
I will say that the term “core” is not synonymous with the term abdominals. So care has to be used to determine what definition
Likewise there is a difference between entering a pose and being in it.
Take Sirsasana for example. Depending on who you are listening to there may be great demand on “core” muscles. Frankly that word to me is a four letter word and I avoid it like the plague whenever possible.
Being in the pose however requires a certain alignment and balance that does not mandate abdominal intensity.
If we shift the focus to Sarvangasana, again the entry does demand some abdominal muscle enlistment but being in the pose actually mandates softening the contents of the abdominal cavity.
But I don’t prescribe exercises as preps. We do preparatory things for each of the inversions we’re heading toward depending on their requisite actions and openings.