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From the Immigration Rules Appendix V https://www.gov.uk/guidance/immigration-rules/immigration-rules-appendix-v-visitor-rules
How long can a visitor extend their stay in the UK
V 8.7 A visitor (standard) and a visitor for marriage or civil partnership, who was granted a visit visa or leave to enter for less than 6 months may be granted an extension of stay as a visitor so that the total period they can remain in the UK (including both the original grant and the extension of stay) does not exceed 6 months.
If your mother-in-lawβs current planned stay means she will have spent 4.5 months in the UK, she can apply to stay for up to a further 1.5 months.
The application must be submitted online and whilst she is in the UK. The fee is Β£993 plus Β£19.20 to provide biometrics https://www.gov.uk/standard-visitor-visa/extend-your-visa