Is it permissible for a woman to deliver lessons in mosques? And what is the ruling on designating a mosque exclusively for women?

...This is a dangerous initiative and a bad practice that did not exist before. Do not say that women need it, or whatever else. We have heard this for a long time from many practicing Muslim women, some of whom are Salafi and some of whom are Khalafi. As we know for certain, whenever a religious innovation is introduced, a recommended practice is abandoned. I saw with my own eyes in the Salah al-Din Mosque, where our brother Muhammad Shuqara delivers the sermon. I went for a prayer, I do not remember now if it was the midday or the afternoon prayer, and I saw some men entering the mosque, meaning the area that is now the sacred precinct from the courtyard, and then they returned. When I entered and wanted to go in, seeing that the men were turning back, I also turned back. Better to see, right? They said the mosque was completely occupied by women, and the teacher was there in the front or on the pulpit teaching. So they occupied the mosque and prevented the men from entering to pray.

The questioner : During the time of prayer.

The Shaykh : No, it appears from what I have heard that Abu Malik, meaning, with difficulty he was able to get out, because the women who were present were taking up a large portion of the mosque. Then the women increase in number, meaning they keep entering. In any case, I imagine now, exactly as it came in some hadiths I imagine now that there will be special mosques for women with an imam who leads the women in prayer.

The questioner : Currently, this exists in Zarqa... a school.

The Shaykh : No, no, I mean a mosque, not a mosque in a school. That is a prayer place. That is a prayer place.

The questioner : A regular mosque with an external door, having an imam, and the call to prayer is made in it... the congregation of the Husayn Mosque...

The Shaykh : No, the mosque is that a mosque is established.

The questioner : It is the establishment of a mosque.

The questioner : The last mosque, for whom?

The questioner : For the student in the school.

The Shaykh : My brother, it is not our concern. In any case, I see that in the distant future, there will be special mosques for women. And this is undoubtedly contrary to the Sunnah. Her prayer in her home is better for her than her prayer in the mosque. The cause of all this is heedlessness regarding the guidance of the Messenger (peace and blessings be upon him). And the cause of heedlessness regarding the guidance of the Messenger is ignorance of the Sunnah. Because the one who wants to say that something is an innovation or a Sunnah needs to have a very comprehensive knowledge of the noble Prophetic life that he (peace and blessings be upon him) lived in his time with his Companions, both women and men. Only then can he say that this is a Sunnah and this is an innovation.