Who is obliged to fast sixty consecutive days? If it is obligatory for a woman, how can she fast sixty consecutive days although she will have menstruation periods? Please, answer me in details about the rulings that make such expiation obligatory, especially those concerned with women?

All perfect praise be to Allah, The Lord of the Worlds. I testify that none is worthy of worship except Allah and that Muhammad is His slave and messenger.

Fasting two consecutive months as expiation is obligatory in three cases:

Firstly, killing by mistake. If a person kills another by mistake and does not find a slave to free, then he is required to fast for two consecutive months. This is based on the statement of Allah, The Almighty (which means): {And never is it for a believer to kill a believer except by mistake. And whoever kills a believer by mistake—then the freeing of a believing slave and a compensation payment presented to the deceased's family [is required] unless they give [up their right as] charity. But if the deceased was from a people at war with you and he was a believer—then [only] the freeing of a believing slave; and if he was from a people with whom you have a treaty—then a compensation payment presented to his family and the freeing of a believing slave. And whoever does not find [one or cannot afford to buy one]—then [instead], a fast for two months consecutively, [seeking] acceptance of repentance from Allah. And Allah is ever Knowing and Wise.} [Quran 4: 92]

Secondly, cases of thihaar, where someone abstains from having sexual intercourse with his wife, claiming that she is like (the back of) his mother. Whoever does so and fails to find a slave to free must fast for two consecutive months as expiation for that. Allah, The Almighty, says (what means): {And those who pronounce thihaar from their wives and then [wish to] go back on what they said—then [there must be] the freeing of a slave before they touch one another. That is what you are admonished thereby; and Allah is Acquainted with what you do. And he who does not find [a slave]—then a fast for two months consecutively…} [Quran 58: 3-4] 

Thirdly, having sexual intercourse with one’s wife during daytime in Ramadan. Thus, whoever does so and cannot find a slave to free must fast for two consecutive months.

When someone is obligated to fast two consecutive months, he  must not interrupt the fasting, except for a valid excuse, such as illness, madness, or menstruation. If one interrupts it for an invalid excuse, then he is obligated to re-start the fast from the beginning. However, if he breaks the fast for a sharia-approved reason, then he does not start anew but should resume from where he stopped fasting once the excuse no longer exists. He should complete what he had already fasted before having such an excuse. In this way, it is clear that if a woman is required to fast two consecutive months and then has her period, she should continue on the basis of the fasts that she has observed before having her period.

It is worth noting here that a woman is not obligated to offer expiation, according to the preponderant opinion, for having sexual intercourse during daytime in Ramadan.  A woman is also not obligated to fast as expiation for thihaar, according to the consensus of Muslim scholars.

Allah Knows best.