All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
These actions that you attributed to your sister are among the ugliest sins and reasons for incurring the Wrath of Allah, The Almighty. If it is true that she had committed these sins, then she is in great danger. You did right by threatening her to inform your mother of her actions. You should monitor her and continue advising her and reminding her of Allah, The Exalted. If she repents and adheres to righteousness, then you should conceal her sins, and this is the required action in this case. It was narrated on the authority of ‘Abdullah ibn ‘Umar that the Prophet said: “A Muslim is a brother of his fellow Muslim; he should not oppress him, nor let him down. Whoever fulfills the needs of his brother, Allah will fulfill his needs, and whoever relieves his Muslim brother of a distress, Allah will relieve him of one of the distresses of the Day of Judgment, and whoever conceals the faults of a Muslim, Allah will conceal his faults on the Day of Judgment.” [Al-Bukhaari and Muslim]
If she is still committing these sins, then you should do everything within your power to deter her and prevent her from doing these heinous acts, whether by informing your mother or otherwise, but without publicly disgracing her. The Maaliki scholar Al-Mawwaaq said in At-Taaj wa Al-Ikleel: “If a woman keeps the company of corrupt people for a while and then comes (of her own accord) to confess it or is driven by someone to do so, the Muslim ruler has no right to investigate whether or not she had committed Zina. He should rather discipline her without exposing her faults. Al-Burzuli said: ‘This is because the objective of the Sharee‘ah is the concealment of sins, as mirrored in the statement of the Prophet when he said (to the man who advised an adulterer to go to the Prophet, sallallaahu ‘alayhi wa sallam to confess his sin): ‘If you had covered him with your garment, it would have been better for you,’ and (when he said to the man who came admitting that he had committed adultery): ‘Perhaps you only kissed; perhaps you only touched…’” [End of quote]
We would like to remind you of the importance of supplicating Allah, The Exalted, to bless her with repentance and guidance, as Allah is All-Hearing and Ever-Responsive. He Says (what means): {And when My servants ask you [O Muhammad] concerning Me – indeed I am near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be [rightly] guided.} [Quran 2:186]
We would also like to point out the gravity of living in non-Muslim countries where Muslims face many dangers, especially for children due to the access to unrestricted freedoms, temptations, and (gratification of sinful) lusts. This is why the scholars warned against living in non-Muslim countries for no necessity or need and believed that it is obligatory on a Muslim who fears falling into Fitnah (temptation) to migrate to a Muslim country.
Allah Knows best.