Encryption doesn't guarantee that the message can't be tampered with, though. In practice I believe it would be quite hard to tamper with the inner signature if you didn't have the encryption key, but relying on encryption to save you from tampering is using the wrong tool for the job. That's the whole reason you have signatures in the first place, otherwise you could just encrypt the plaintext and send it and have the fact that you knew the right key provide authentication.
Note that in the context of encryption, signatures for authentication don't mean public/private signature schemes, but just a plain MAC using a shared secret. Authentication in the sense of "Do I know I'm talking to who I think I'm talking to?" is handled at a separate level as part of the initial key exchange. For an authenticated encryption scheme, you'd exchange both the encryption key and the MAC key as part of that exchange. There's no sensible scenario where an attacker would know one of those keys and not the other, because they're generated and stored together. In fact, as far as anyone knows there's nothing wrong with using the same key for both, it's just that nobody is completely sure that's safe. Since it's easy to generate and use two keys instead, that's recommended.
Note that in the context of encryption, signatures for authentication don't mean public/private signature schemes, but just a plain MAC using a shared secret. Authentication in the sense of "Do I know I'm talking to who I think I'm talking to?" is handled at a separate level as part of the initial key exchange. For an authenticated encryption scheme, you'd exchange both the encryption key and the MAC key as part of that exchange. There's no sensible scenario where an attacker would know one of those keys and not the other, because they're generated and stored together. In fact, as far as anyone knows there's nothing wrong with using the same key for both, it's just that nobody is completely sure that's safe. Since it's easy to generate and use two keys instead, that's recommended.