Everyone agrees that an abortion beyond a certain gestational age is forbidden for a Jew 1. However, there's no death penalty for it. This is unlike a non-Jew, who is liable to capital punishment for destroying a fetus 2 (Tosafos to Sanhedrin 59a s.v. ליכא, quoting Sanhedrin 57b).
However, there are certain circumstances where an abortion is permissible or mandatory. For example, if the fetus is a lethal threat to the mother (Oholos 7:6). Sometimes there are other considerations as well, and a person should always ask their competent halachic authority.
What I'm wondering is, in the situation that a abortion is permissible (or even mandatory), who should perform it? A Jewish doctor, and not a non-Jewish one (since for them it's seemingly a more serious transgression)? Or even a non-Jewish doctor can? Is it better for the mother to do it herself, it at all possible? Does it make a difference? Does it matter if the reason the abortion is permissible is because of a threat to life (in which case the doctor should be allowed), or for other reasons (in which case, why should the doctor be allowed if the mother could)?
I'm not so familiar with how they do abortions. I would imagine in certain cases there's simply a pill to take (early in the pregnancy), and later in the pregnancy a procedure needs to be performed. I'm more wondering the latter. Where a procedure needs to be performed, who should do it?
For a potentially related discussion: See Imrei Moshe § 12, who discusses the prohibition of cutting off one's peyos. If for medical reasons someone has to have them cut off, he wonders if it's better to have a non-Jew do it, since they aren't prohibited in cutting off someone's peyos, rather than a Jew, who is prohibited from cutting. He then innovates that if the person themself is allowed to have their peyos cut off, someone else wouldn't have the prohibition of cutting it. That may or may not be helpful here.
1 There are different suggestions for what the prohibition is. See this answer to a related question.
2 See this related question.