In Nov 1960, Krypto appeared in three stories, and this being the very heart of the Silver Age, all three are about faking out the characters and/or the readership. And all three feature Supergirl, oddly enough.
In "The Old Man of Metropolis", Superman reads Supergirl's essay about one day becoming Superwoman and dreams of skipping through life and time and finding himself a forgotten, decrepit, powerless old man in the future she imagined. Krypto is there too, in the same state.
Supergirl has another shot at saving Krypto from a "pound" in the next story, "Supergirl's Busiest Day!", during which she comes across, among other things, a flying saucer grabbing alien animals, including our dear old Krypto.
That's her agony in the Adventure story, "Supergirl in Smallville" in which she goes back in time to Superman's boyhood to prove she can be as trusted as he was at the same age. I mean, COME ON! She lets the Kents in on her secret, and Clark is led to believe she's one of their distant relations, staying on a perhaps permanent basis. She does a lot of good for the community, saves his bacon, fixes his robots when he's not looking, etc., but when she sees Krypto, whom she's met, angling for Smallville, she can't let him recognize her!
But helping people and confusing dogs is one thing. Can she pass the ultimate test? Can she hide her identity from nosy Lana Lang trying to trick anyone and everyone into outing themselves as Kryptonians? Not without some help from the Kents and Krypto!
And Krypto's complicit!