I have just tried to install the current version of the Android player onto a Lenovo Yoga Tab 3 (android 5.1), and a Samsung tablet (android 4.4).
I have ensure that the installation from unknown source is enabled, but I cannot install on both of them.
On the Lenovo Yoga Tab 3, the APK cannot open. And on the Samsung, it opens, and shows the required permission, but the INSTALL button cannot be clicked.
You may need to use a file manager such as Astro or ES to open the file on your Lenovo. Some manufacturers prevent apks from being launched from the Downloads in the browser directly, so that should make it work there.
On your Samsung, perhaps you have to scroll through the list of permissions before you can click install? If not, you may need to contact Samsung directly as the APK installation is part of the tablet’s software, not something that we include in the APK.