Praise be to Allah.
Revocable divorce is the first or second talaaq, so long as it is not talaaq in return for money (i.e., so long as it is not khula‘). Then this revocable divorce is counted as one of the three talaaqs, and it is permissible to take the wife back during her ‘iddah, even if that is without her consent, with no new marriage contract or mahr; rather the original marriage contract remains in effect.
As for irrevocable divorce (at-talaaq al-baa’in), it is divided into two types, major and minor. Irrevocable divorce in the major sense means that the husband has completed three talaaqs, as a result of which the marriage contract between them comes to an end, and it is not permissible for him to marry her again until after she has married another husband in a genuine marriage, not a tahleel marriage (one aimed making it permissible for her to go back to the first husband), then the second husband divorces her or passes away.
Irrevocable divorce in the minor sense refers to cases where talaaq is issued in return for compensation, or it occurs before consummation of the marriage, or where the husband has issued one or two talaaqs and has not taken her back before her ‘iddah ended. As a consequence of that, the marriage contract between them comes to an end, but it is possible for the husband to remarry her with her consent, and with a new marriage contract and mahr. This has been explained previously in fatwa no. 46561.
Hence it is known that this talaaq that your friend signed for his wife is a revocable talaaq, so long as it did not happen in return for money, and was not a third talaaq, as appears to be the case from your question.
The ‘iddah of a woman divorced by talaaq varies according to her situation. This has been explained previously in detail in fatwa no. 12667
So long as this woman menstruates, as is mentioned in the question, then her ‘iddah is according to her menstrual cycle. So what she must do is wait until she has had three menstrual cycles after her divorce, including the menses that came on the day of the talaaq, if he divorced her before her menses began, as we shall see below. If her husband took her back during this ‘iddah, i.e., before her third menses, then she is still his wife.
But if he took her back after the end of the ‘iddah, then in that case the taking back is not valid, and she has become irrevocable divorce in the minor sense.
The fuqaha’ are unanimously agreed that taking back the wife is valid if done by means of words to that effect, such as if he says to her: “I take you back” or “I take her back”. However they differed as to whether taking back may be achieved by means of actions, such as intercourse and the things that lead to it, e.g., touching, kissing, being alone together. This has been discussed previously in the answer to question no. 23269.
You say that your friend sent her a text message saying that he was taking her back. If he intended to take her back by means of this message, then it was fulfilled thereby. The scholars have stated that writing down the taking back, with the intention of doing so, counts as taking back. It says in I‘aanat al-Taalibeen ‘ala Hall Alfaaz Fath al-Ma‘een (4/34): Writing may take the place of speaking, if accompanied by the intention. End quote.
However we should point out that just as it is not necessary to have the wife’s consent to taking her back, it is also not necessary to inform her of it; rather even if he takes her back without her knowledge, the taking back is still valid, and she is obliged to comply with the implications of his taking her back once she is certain of his claim to have taken her back during the ‘iddah, or once he establishes proof for her.
It says in Badaa’i‘ as-Sanaa’i‘ fi Tarteeb ash-Sharaa’i‘ (3/181):
Similarly, informing her that he is taking her back is not an essential condition; even if he does not tell her that he is taking her back, that is acceptable, because taking back is his right exclusively. End quote.
Issuing a talaaq during menses is a sin and an act of disobedience; it is contrary to the command of the Prophet (blessings and peace of Allah be upon him). As to whether it is applicable to the wife, that is a matter concerning which the scholars differed. We have discussed this difference of opinion previously; for information on the most correct view, please see fatwa no. 72417
This difference of opinion is of concern if the husband issued a divorce to his wife at the time when she was menstruating. As for completing the official procedures and filing the divorce papers, that has nothing to do with whether the woman is menstruating or not. If he divorced her during a period of purity in which he had not had intercourse with her, but the divorce papers were signed when she was menstruating, then this divorce was done in accordance with the Sunnah and counts as such, according to scholarly consensus, even if she got her menses on the same day, after he had actually divorced her.
And Allah knows best.