Even with a verified APK, Android 4.2.2 is a dead platform. Newer apps will increasingly require Google Play Services versions that are incompatible with Jelly Bean. You may find that after installing the legacy APK, some apps still fail. In this case, your best long-term solutions are:
Even if a "verified," unmodified Google Play Services APK is successfully installed on Android 4.2, the device remains insecure.
To avoid the dreaded “App not installed” error, download the correct variant.
For 99% of Android 4.2.2 devices, the safe choice is
armeabi-v7a + nodpi.
Since Android 4.2.2 is obsolete, you cannot update Play Services directly through the Play Store interface if it is broken or missing. You must use a trusted third-party APK repository.
Recommended Verified Sources:
A: Android 4.2.2 had a separate /data partition for system apps. Try moving some user apps to SD card, then install Play Services again.