Praise be to Allah.
There are several problems with artificial insemination, and taking this matter lightly leads to serious negative consequences. It is essential for both partners to be cautious with regard to this matter and to ensure that this procedure is done by trustworthy medical professionals.
Please see the answer to question no.3474
The scholars differed concerning taking back the wife when her ‘iddah has not yet ended: is that accomplished just by doing the deed, or is it essential to speak words to that effect, as at the beginning of marriage?
It says in al-Mawsoo‘ah al-Fiqhiyyah (22/9-12):
The fuqaha’ are unanimously agreed that taking back the wife is valid if words are uttered to that effect, such as if he says to his (revocably) divorced wife before the ‘iddah ends, “I take you back” or any other words to that effect.
With regard to taking back the wife by means of actions:
The Hanafis are of the view that intercourse and things that precede it are valid for taking the wife back. The Maalikis are of the view that taking the wife back is valid in the form of actions such as intercourse and things that precede it, on condition that the husband intends by these actions to take his wife back.
According to the Shaafa‘is, taking back the wife in the form of actions is not valid in any circumstances, whether it is done by intercourse or the things that precede it, and whether or not the action is accompanied by the husband’s intention to take her back.
The Hanbalis differentiated between intercourse and the things that precede it, with regard to the validity of taking the wife back. In their view, taking the wife back in the form of intercourse is valid, but not in the form of things that precede intercourse.
The most correct scholarly opinion concerning this matter is that taking the wife back may be done by actions, if the intention is to take her back. However, if he does not intend to take her back, then that cannot be done merely by actions.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
He may take her back by uttering words, so he may say: I take my wife back. Or he may take her back by means of actions, so he may have intercourse with her with the intention of taking her back.
End quote from Fataawa Islamiyyah, 3/423
See also the answer to question no. 101702
Based on the above, if the husband intends to take his wife back, then it is permissible for them to undergo artificial insemination, and it is sufficient for them to go ahead with this procedure, if the husband intends thereby to take his wife back. It is not stipulated that he should take her back by uttering words, although that is preferable and more on the safe side.
If he has taken his wife back by his words or by his actions accompanied by the intention, then her ‘iddah has ended, because she is no longer regarded as divorced; rather she has gone back to being truly married. So there is no need to consider her ‘iddah or how it will end, unless he divorces her again.
But if he did not intend to take his wife back, it is not permissible for him to go ahead with artificial insemination with her when she is divorced from him, because artificial insemination is only permissible with one’s wife, and merely doing artificial insemination without the intention of taking her back does not constitute taking her back.
And Allah knows best.