The Extradition Act deals with extradition in India.
But in its Sections it doesn't have any such provisions or details. and can State High court of India enquire into such extraditions?
For example, the extradition treaty between India and Türkiye specifies in its Article 8(1) that
The nationals of a Contracting State shall not be extradited to the other Contracting State provided that the Requested State shall submit the case to its competent authorities for prosecution if the act committed is also considered as an offence under the laws of the Requested State.
That is, Indian citizens in India would not be extradited to Türkiye, but instead could be prosecuted in India. Provisions of the Extradition Act mean that the offence is treated as if it had occurred in India, for such a prosecution. This does not stop an Indian citizen in Türkiye from being extradited to India, or the reverse situation.
On the other hand, the agreement between India and the United States says in Article 3 that
Extradition shall not be refused on the ground that the person sought is a national of the Requested State.
which alongside the general obligation in Article 1 means that Indian citizens in India could be extradited to the USA, assuming all other conditions are met.
Regarding the High Court question: While the extradition proceedings must take place before a magistrate, they can be challenged by way of judicial review. The same applies to decisions made by central government as part of the process. In India this is called "writ jurisdiction" and can be exercised before a High Court, or the Supreme Court. It is limited in scope and usually would only apply if there has been some procedural irregularity, irrationality in decision-making, or violation of fundamental rights.
In answer to the title's question: Yes
And an example of an Indian citizen being extradited to India is Sanjeev Chawla which was India’s first successful extradition since the treaty was signed with the UK in 1992
Also, for interest and further reading, this Motaq article gives some background regarding India - UK extradition.