There are two Type of Automation (in this specific Scenario)
Unit Test Automation, in which we develop automation scripts for Unit testing. I have seen project where Jmeter and other tools are used and the automation testing teams doing this automation with the help of Developers.
Function Test Automation, in which Automation team gets the product / service with test cases (Waterfall model not agile that is different story) and do the automation using specific automation testing tools.
Now For your precise question "Who Should Do the Automation Test ? " (in which you have not cleared what automation) Answer is Test Automation team should do the automation.
If you ask any of the following questions then answer(s) will be different
Who should do the Unit Test Automation ?
Who should do the Load Automation Test ?
Who should do the non function Automation test ?
Who should do the Black box test automation ? (In other words Functional automation)
Then There are several (read hundreds) other test automation areas.