She claims that there is no evidence that a woman who menstruates has to make up missed fasts

The fact that a woman who menstruates is obliged to make up missed fasts is a ruling on which there is consensus among the Muslims, and is indicated by the saheeh Sunnah and scholarly consensus. 

Al-Bukhaari (321) and Muslim (335) narrated that Mu’aadhah said: I asked ‘Aa’ishah: “Why does a woman who menstruates have to make up the fasts but not the prayers?” She said: “Are you a Haroori?” I said: “I am not a Haroori, but I am asking.” She said: “That used to happen to us and we were commanded to make up the fasts but we were not commanded to make up the prayers.” 

Al-Nawawi (may Allaah have mercy on him) said: 

This ruling is agreed upon. The Muslims are unanimously agreed that menstruating women and women who bleed following childbirth are not obliged to pray or fast at that time, and they are agreed that they do not have to make up the prayers, and they are agreed that they have to make up the fasts. The scholars said: the difference between them is that the prayers are many and repeated, and it would be too difficult to make them up, unlike the fast which is enjoined only once in the year, and a woman’s period may last only one or two days. 

Al-Haafiz said: 

“Are you a Haroori?” – the word Haroori is derived from Haroora’, which is a village two miles from Kufa. Those who believe in the madhhab of the Khaarijis are called Harooris because the first group of them rebelled against ‘Ali in that village, so they became known by this name. There are many such groups, but one of their basic principles on which they are all agreed is following what is indicated by the Qur’aan and rejecting additional evidence from the hadeeth. Hence ‘Aa’ishah asked Mu’aadhah this question by way of criticism.  

Ibn Qudaamah said in al-Mughni (3/39): 

The scholars are unanimously agreed that it is not permissible for menstruating women and women who bleed following childbirth to fast, and that they should not fast in Ramadaan, and should make up the missed days, and that if they do fast their fast does not count. 

Al-Nawawi said in al-Majmoo’ (2/386): 

The ummah is also unanimously agreed that it is obligatory to make up missed Ramadaan fasts. Al-Tirmidhi, Ibn al-Mundhir, Ibn Jareer, our companions and others narrated that there was consensus on this point. 

Shaykh al-Islam said in Majmoo’ al-Fataawa (25/219): 

It is proven by the Sunnah and the consensus of the Muslims that menstrual bleeding invalidates the fast, so a menstruating woman should not fast, but she should make up the missed fast. 

This is the Sunnah of the Prophet (peace and blessings of Allaah be upon him), and scholarly consensus on this matter was narrated by more than one of the scholars.  

So how can anyone then say that there is no evidence that a woman who menstruates has to make up missed fasts! 

This woman referred to in the question has to repent to Allaah from this wrong idea which implies rejection of the laws and rulings of Allaah. Whoever does not know something has to research and ask the scholars; it is not permissible for him to speak about the religion of Allaah without knowledge, because that is something that is forbidden. Allaah says (interpretation of the meaning): 

“Say (O Muhammad): (But) the things that my Lord has indeed forbidden are Al-Fawaahish (great evil sins and every kind of unlawful sexual intercourse) whether committed openly or secretly, sins (of all kinds), unrighteous oppression, joining partners (in worship) with Allaah for which He has given no authority, and saying things about Allaah of which you have no knowledge”

[al-A’raaf 7:33]

The Muslim should realize that he is responsible for every word he says. Allaah says (interpretation of the meaning): 

“Not a word does he (or she) utter but there is a watcher by him ready (to record it)”

[Qaaf 50:18]

We ask Allaah to bless us with insight and understanding of His religion. 

And Allaah knows best.