Unless the employee is actually in the HR department, HR should not actually be the ones making the decision about whether or not to fire an employee.
The documents you need to look at in your business are the contract of employment and the employee handbook. It is essential that, if you have these documents, you stick to your company policy as laid out in the documents. Failure to do so may lead to an unfair dismissal, regardless of the events leading up to the dismissal (firing). Do not put yourself in breach of your own policies and procedures.
In most cases, HR are in a supporting role. They may carry out the investigation, they may chair a disciplinary meeting or attend as a note taker, but the actual decision on whether or not to terminate someone’s employment should rest with the line manager.
Another role which HR typically plays in these situations is advisor, especially if your HR support is external to the business. It may be that HR ensures the employee has been given their full rights, that the process being followed is legal or that the manager carrying out the disciplinary is properly trained to do so.
If you are working in a larger business with an HR department and the employee being terminated is in HR, then yes, HR can fire the person; but in most other situations, their role is advisory and support.