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Puppet Stayman for 2NT is similar with the Puppet Stayman for 1NT. Also it can be used for all variation of 2NT Strong sequence. (Read more in the NT core-structure) Whether you bid the opening 2NT directly, via Multi-2D or by bid 2♣ first, as long as your balance strong bid can contain a 5-cards-major suit, then “Puppet Stayman” is suitable for the job.
Notes:

2NT-3♣ = Puppet Stayman: Opener’s Re-Bid:

…
2NT – 3♣ (Puppet Stayman)
?
3♦ : No 5-cards in Major, but I have one or two 4-cards in Major
3♥ : I have 5-cards Heart
3♠ : I have 5-cards Spade
3NT : No 5-cards major nor 4-card major

Subsequent Bid

After opener rebid 3♦

The responder bid 4-cards Major that the responder has, in “transfer manner”. Meaning, bid the major that the responder did NOT have. The purpose if to make sure the stronger hand become the declarer.

…
2NT – 3♣ (Puppet Stayman)
3♦ – ?? (No 5-cards in Major, but opener has one or two 4-cards in Major)
– 3♥ : I have 4-cards Spade (bid what you do NOT have). If fit, opener then bid the major. If not, bid 3NT.
– 3♠ : I have 4-cards Heart (bid what you do NOT have). If fit, opener then bid the major. If not, bid 3NT.
– 3NT : Sign off

After opener rebid 3-Major

Once the opener bid 3♥ or 3♠, the responder know 11 cards at opener hand(5 in major and 2 each in each other suit). Therefore, all jump bids are “auto-fit”. And because the level is already one level higher than 1NT sequence, there is no splinter or cuebid sequence. However the exclusion key card will still be available (because of the strength in point, exclusion key card is more likely to be useful). The cuebidding can still be done after RKC.

…
2NT – 3♣ (Puppet Stayman)
3♥ – ? (Opener has 5-cards Heart)
– 3♠ : Natural, 4+cards
– 3NT : Invite balance (no fit)
– 4♣ : Natural, 4+cards (no fit)
– 4♦ : Natural, 4+cards (no fit)
– 4♥ : Sign off
– 3♠ : Auto-fit. Exclusion key card in ♠
– 4NT : Auto-fit. RKC in ♥
– 5♣ : Auto-fit. Exclusion key card in ♣
– 5♦ : Auto-fit. Exclusion key card in ♦
…
2NT – 3♣ (Puppet Stayman)
3♠ – ? (Opener has 5-cards Spade)
– 3NT : Sign-off
– 4♣ : Natural, 4+cards (no fit)
– 4♦ : Natural, 4+cards (no fit)
– 4♥ : Natural, 4+cards (no fit)
– 4♠ : Sign Off
– 4NT : Auto-fit. RKC in ♠
– 5♣ : Auto-fit. Exclusion key card in ♣
– 5♦ : Auto-fit. Exclusion key card in ♦
– 5♥ : Auto-fit. Exclusion key card in ♥

After Opener rebid 3NT

If the opener re-bid 2NT, then by default, the opener has

  • At least a 3-2 in the Major suit
  • At least a 4-3 in the minor suit.

So, in a way, responder (and opponents!) know of at least 12 cards of opener’s hand.

Using this knowledge, the responder should be able to decide the final contract. 4NT = Quantitative. 4♣= Gerber

Defensive Bidding for Puppet Stayman

  • Upon opponent’s double: all system on (i.e ignore the Double)
  • Upon opponent’s 3♣ overcall:
    • If the 3♣ is artificial: then double is a “replacement bid” (as if you bid that 3♣). In other words, Puppet Stayman is still on
    • If the 3♣ is natural: Cannot use Stayman. DBL = Penalty
  • If opponent bid higher than 3♣: also you cannot use Puppet Stayman anymore.

Competitive Bidding for Puppet Stayman

If the opponent is using this convention, after 3♣ Stayman by the opponent:

  • Use DOUBLE as “lead-directing double” (have Clubs)
  • Other overcall is natural

Bridge Link: European Bridge League

[Puppet Stayman Convention for 2NT]