Is it permissible to write in her resume that she has experience that she did not acquire in the manner required?
The Muslim should speak the truth and seek to be honest. Allah, may He be exalted, says (interpretation of the meaning):
“O you who believe! Fear Allaah, and be with those who are true (in words and deeds)”
Al-Bukhaari (6094) and Muslim (2607) narrated that Ibn Mas‘ood (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “I enjoin you to be truthful, for truthfulness leads to righteousness and righteousness leads to Paradise. A man may continue to tell the truth and endeavour to be truthful until he is recorded with Allaah as a speaker of truth. And beware of lying, for lying leads to wickedness and wickedness leads to Hell. A man may continue to tell lies and endeavour to tell lies, until he is recorded with Allaah as a liar.”
al-Nawawi (may Allah have mercy on him) said:
The scholars said: What this means is that truthfulness leads to righteous deeds that are free of everything blameworthy. Al-birr (righteousness) is a general word that includes all kinds of good deeds. But lying leads to wickedness, which is drifting away from righteousness or, it was said, indulging in sin.
The scholars said: This is encouragement to endeavour to be truthful, which means seeking it and paying attention to it. (And it is encouragement) to beware of lying and taking the matter lightly, for if one takes the matter lightly he will do a great deal of it and become known for that. So if a person tells the truth all the time and it becomes second nature to him, Allah will record him as a speaker of truth. On the other hand, if he tells lies all the time and it becomes second nature to him, Allah will record him as a liar. End quote.
Mentioning your teaching at home as if it were teaching in an educational institution is undoubtedly a kind of lying. Rather you have to tell the truth and tell them that you taught privately at home.
Remember that if you tell the truth and fear Allah, Allah will provide for you from sources you could not imagine. Allah, may He be exalted, says (interpretation of the meaning):
“And whosoever fears Allaah and keeps his duty to Him, He will make a way for him to get out (from every difficulty).
3. And He will provide him from (sources) he never could imagine”
And Allah knows best.