Why did you give the wrong details, and what information was incorrect

Your insurer can sue you in certain circumstances - for costs of the claim they've had to pay out

No but if you do so you will be in contravention of Road Traffic Act and liable for prosecution.

Deliberately giving false information to avoid prosecution is attempting to pervert the course of justice.

Can my insurance company sue me for giving wrong details to the other driver for an accident I was at fault for.

What exactly happened?

Did you make an honest mistake or give information knowing it to be wrong?