Without giving it too much importance, he keeps the object in his pocket, without releasing it at any time.
Naruto was headed home at first so maybe his mother has some idea about what the amulet is or what it's for.
Arriving home, Naruto walks towards the kitchen looking for the Uzumaki woman, and finding her with her back turned doing housework, he takes the amulet out of his pocket, at the same time catching her attention. Just then, out of nowhere, the amulet begins to glow mysteriously...