Never give your private key! Once you give it, they can enter your wallet and take all the money. And that transaction cannot be reverted, so you will lose your money.
Looks like fraud
Looks like ? It is absolutely fraud. You have said it all and there is nothing they need the private key for than to have access to the full wallet, so the OP should better just know that now and try not to ever make a mistake like that. Also, be careful the link you open directly from your mails or anywhere as those crazy thieves are on the loose, looking for every account to pounce on.
We should learn to treat our private key like our own eyeball. Even if some people are doing airdrop, the only thing they need from us is our public address and if we ever fall for something like this, it may not even be Ethereum, it could be other asset, then we will have ourselves to blame. It is good that OP have asked as lack of information has made people to learn the hard way.