Does a woman who prays Fajr at home and on time receive the reward of having prayed the entire night, as stated in the Hadith: "Whoever prays 'Isha in congregation, it is as if he has prayed half the night. And whoever prays Fajr in congregation, it is as if he has prayed the entire night"?

The hadith mentioned in the question is authentic, and whoever does not pray in congregation will apparently not attain the promised reward. However, it is hoped that Allah will not let the reward of their good intention be lost. As for a woman who prays in congregation in her home, it is hoped that she will achieve this reward, because it was not stipulated that the congregation must be in the mosque. A woman's prayer in her home is more appropriate and better, due to the modesty and avoidance of temptations it offers. So, if they pray in congregation in their homes, they combine avoiding temptations with attaining the reward of congregational prayer.