What to do about my cat?

Nothing you can do; cats are hunters by nature. Those lizards aren't so innocent, they kill and eat bugs as a large part of their diet.

When dealing with our pets the most important thing to remember is that they are animals. They're not little furry people, and that applies to all aspects of pet ownership.

One thing I found somewhat successful is not reacting to the 'gift'. Even if you were to scold the cat it's still attention. And like people, even negative attention is at least attention. It's also helpful to make sure the cat is well entertained. They don't make the distinction between a living thing and their toys. Cats are surprisingly intelligent and need to exercise their minds as well as their claws.

But the instinct to hunt is so deeply ingrained in the animal that there's little anyone can do to change it. We have to recognize and respect that our loving little fluff ball is, by nature, a predator. It's built to hunt, kill, disembowel, and eat its prey. We can easily change some behaviors. Even an outdoor cat will quickly know to use a litterbox. No one is sure why, but they recognize it immediately and know what it's for. But they also recognize prey animals and know what they're for as well. Your cat is the pointy little tip of the food chain in your yard and they are made to fill that role in nature..

It is seemingly strange that domestic cats will hunt even if they don't need food. They are one of the rare species that will do that. But it's in their nature and we need to respect that and accept the whole animal, for good or bad.

Difficult to fight that instinct... some cats just can't help it, and for her it's a present she's bringing for you... Even if they could have an awareness for other animals' lives or if she knew you don't like them to bring you their prey, they would find it really difficult to resist so they'd still do it behind your back. Some (but fewer) dogs have the same problem, and unless these animals have been raised among other smaller species and learned to live in friendship with them (and sometimes even with that they might still hunt a "stranger", even if they treat a pet lizard/mouse/bird that lives with them as a friend).

Keep her inside.

Kittys stalk, hunt and kill instinctively, whether or not they are hungry.

Besides, in your kitty's mind she is helping YOU, by bringing YOU food back to the communal lair. It is a high compliment and sign of her high regard for you.

Outdoor cats, whether feral or not, are responsible for an incredible amount of wildlife destruction. They do not know the difference between rodent pests and endangered songbirds or lizards.

Cats should be kept inside.

There is nothing you can do except perhaps post this in the pet category since your cat catching prey has NOTHING to do with the food and cooking category which is for humans.

Cats, like human primates, sometimes kill for sport/fun/convenience (among other things). When you break it down, nature is just one big sick circle of lifeforms trying to kill/eat one another. There is nothing intelligent or benevolent about that "design".

nothing. cats are natural predators,