You cannot apply for an extension of the visa based on your relationship.
If the relationship is long term, you may think about getting married in which case she would be eligible to make an application for a spouse visa.
The only alternative is for yoru girlfriend to make an application for a visa under the Points Based System, either a Tier 1 or Tier 2 type Visa.
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If you wish to discuss, please feel free to ask further questions.
You can marry in the UK. Your girlfreind will need to first apply for a Certificate of Approval from the Home Office. Once granted, she will be entitled to marry. Thereafter, she will be eligible to make an application for a Spouse Visa.
If she is granted a Spouse Visa, she will be allowed to stay in the UK for a period of 2 years. At the end of the 2 years, she will have to apply for Indefinite Leave to Remain.
If her Visa is due to expire in April, I suggest she apply for a Certificate of Approval as soon as possible.
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