install apk from url

This won't help if the app is not available on the mearketplace, but in case it is:

Uri marketUri = Uri.parse("market://search?q=pname:com.appmaker.tefloukipackage");
Intent marketIntent = new Intent(Intent.ACTION_VIEW, marketUri);
try {
    context.startActivity(marketIntent);
} catch (ActivityNotFoundException ex) {
    showAlertDialog(context, "Error", "Could not launch the market application.", true, null);
}

Follow along.

In your module manifest, add

<uses-permission android:name="android.permission.READ_EXTERNAL_STORAGE" />
<uses-permission android:name="android.permission.REQUEST_INSTALL_PACKAGES" />

//Inside application block
<application>
    <provider
        android:name="androidx.core.content.FileProvider"
        android:authorities="${applicationId}.provider"
        android:exported="false"
        android:grantUriPermissions="true">
        <meta-data
            android:name="android.support.FILE_PROVIDER_PATHS"
            android:resource="@xml/provider_path"/>
    </provider>
</application>

In your module res/xml folder, if not create this folder, with file provider_path.xml

<?xml version="1.0" encoding="utf-8"?>
<paths xmlns:android="http://schemas.android.com/apk/res/android">
    <external-path name="external_files" path="."/>
</paths>

And use this method.

private fun updateApplication(activity: Activity) {
    //This will get you the root directory path
    val externalStoragePublicDirectory: String =
        Environment.getExternalStorageDirectory().path

    val externalStoragePublicDirectoryFile =
        File(externalStoragePublicDirectory, "MyApp" + ".apk")

    val uri = FileProvider.getUriForFile(
        activity.applicationContext,
        activity.applicationContext.packageName + ".provider",
        externalStoragePublicDirectoryFile
    )

    val installAppIntent = Intent(Intent.ACTION_VIEW)
        .setFlags(Intent.FLAG_ACTIVITY_CLEAR_TOP or Intent.FLAG_ACTIVITY_NEW_TASK)
        .setDataAndType(
            uri,
            "application/vnd.android.package-archive"
        )
        .addFlags(Intent.FLAG_GRANT_READ_URI_PERMISSION)
        .putExtra(Intent.EXTRA_NOT_UNKNOWN_SOURCE, true)
    activity.startActivity(installAppIntent)
    //This will close your app, remove if not needed
    exitProcess(0)
}

Important, also go to your phone settings, search for unknown sources, enable it in old devices, but in new devices, search for your app and allow it the permission to install new app packages. Only then you will get the pop up to install the app.