Hi. My employer just got a new website ([old site](https://www.hsakft.hu), [new site](https://www.hsa.hu)). The old site has pretty good rankings for our best keywords, but this means that the new site is far behind it.
Should I simply redirect from the old site to the new one? But then the old url will still show up in the search results. Is there a way to let Google know that this is the new official website? Maybe even get better rankings in the process?