Legal Question in Wills and Trusts in Georgia

You can't. Only a person of sound mind can give a power of attorney.

You don't. A person must be mentally competent to execute a power of attorney and the comatose person obviously does not meet that test.

If it is likely that the person will remain in a coma/vegetative state, then you or the family will have to pursue conservatorship/guardianship proceedings. The conservator handles the comatose person's finances and property and the guardian cares for the person - the guardian makes the health care decisions like whether the comatose person should be admitted into the hospital or should have a certain medical procedure.