Lumo TV › Sideload on Firestick
Amazon's app store only shows you what Amazon carries. Everything else installs through a free app called Downloader by AFTVnews, and it takes about five minutes. Nothing here roots or jailbreaks anything. It is a normal, supported Fire OS feature.
From the Fire TV home screen, use the search icon and type Downloader. It is the one named Downloader by AFTVnews, with the orange icon. Other apps with similar names are not the same app. Install it. It is free, and it is in Amazon's own store.
Go to Settings → My Fire TV → Developer options → Install unknown apps, and switch it ON for Downloader.
If you cannot see Developer options at all, that is normal on newer devices: go to Settings → My Fire TV → About, highlight your device name, and press the OK button seven times. A message appears saying you are now a developer, and the menu will be there when you go back.
On a television with Fire TV built in (Amazon Omni, 4-Series, Insignia, Toshiba, Hisense), the same menu lives under Settings → Device & Software instead of My Fire TV.
Open Downloader, click the URL box, and type the address of the APK you want. Downloader will fetch the file and offer to install it.
Many apps publish a short Downloader code instead: a number you type in place of the address, which is far easier with a TV remote. Lumo TV, for example, is 3690365.
Type it carefully. Downloader has no autocorrect, and a single wrong character gives you a "page not found" rather than a helpful error.
Click Install, wait for it to finish, then choose Done rather than Open. Downloader will offer to delete the installer file. Say yes. Fire TV Sticks have very little storage and leftover APKs fill it up fast.
Sideloaded apps do not go on the front row. Press Home, scroll right to Apps, and it will be at the bottom of the grid. Highlight it, press the menu button (the three lines), and choose Move to front so you do not have to hunt for it every time.
"App not installed" usually means the APK is built for a different chip. Fire TV needs an ARM build; an x86 one will fail every time.
It installs but will not open is nearly always a storage problem on the basic Stick. Uninstall something you do not use and try again.
Downloader will not fetch anything means step 2 did not take. Go back and check the toggle is on for Downloader specifically, not just that the menu exists.
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Messages started here include a short code for the page you came from (e.g. LT-GUIDES). It is not unique to you and you can delete it before sending.