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Some, maybe a little helium fusion occurs but it takes much higher temperatures, like 5 or 6 times hotter than the suns core. When the hydrogen burns out the star collapses in on its self, greatly increasing core temperature and pressure until it reignites. Then the star expands way bigger, because it's a lot hotter because when the star reignites on helium it burns a good portion of its helium in a few seconds.
They burn hydrogen for billions of years, helium for millions of years, carbon for centuries, neon for years, oxygen for around a few years or less depending on size, silicon for a day, then collapses to make iron nickel for a few seconds and if it's big enough after iron and nickel it goes boom.