The Responder after a 1 NT overcall
When your Opponent overcalls 1 NT, he has a balanced hand and 15-17HCP. You know immediately the respective strength of the two sides. With 9HCP and more the Responder automatically doubles showing his Partner that they have the majority of the points and is inviting him to let the Opps take the fall. The double is particularly juicy when Opps are vulernable.
The follow-on bids after a 1 NT over call doubled : Opener has a regular distribution.
With a regular distribution, Opener always accepts the punitive invitation. The stronger his opening, the harder the fall !
The follow-on bids after a 1 NT over call doubled : Opener has an irregular distribution
With a very irregular distribution, Opener prefers not to leave the punitive double, but to find a winning contract, especially if his team is vulnerable and Opps are non vulnerable.
The follow-on bids after a 1 NT over call doubled : of course all double bids made by Opener's team are punitive.