If partner's 1NT opening is overcalled with a natural bid then it is generally best to play double as penalties and all other bids as natural (employing Lebensohl).

The one case where I like to differ from this philosophy is when it's a 2♣ overcall.

Over a natural 2♣ overcall I like to play double as Stayman, and other systems are off.

But when the opponent's overcall is artificial (often the case these days) then things are different. For example there is little point in doubling an artificial bid for penalties and little point in bidding Stayman if RHO has shown both majors.

Here is a scheme for combating Cappelletti:

After 2♣ showing a one-suited hand: 

• pass = generally less than invitational values
• double =  Stayman
• 3♣ is natural and forcing
• other bids are basically natural (Lebensohl) 

After 2♦ showing both majors:

• pass = generally less than invitational values
• double = can penalize at least one major
• 3♦ is natural and forcing
• other bids are basically natural (Lebensohl)

After 2♥ showing ♥'s and a minor:

• pass = generally less than invitational values
• double = penalty for the major suit bid
• 3♥ = game forcing Stayman
• other bids are basically natural (Lebensohl)

After 2♠ showing ♠'s and a minor:

• pass = generally less than invitational values
• double = penalty for the major suit bid
• 3♠ = game forcing Stayman
• other bids are basically natural (Lebensohl)

After 2NT (Unusual) showing ♣'s and ♦'s:

• pass = generally less than invitational values
• double = can penalize at least one minor
• 3♣ = Stayman, not game forcing
• 3♦ = Stayman, game forcing
• 3♥/♠ natural and game forcing