I had an issue with this game with input lag, I would hit the jump key and after taking my hand away from the key the action would then happen, which made the game unplayable. After researching this I seen other people had similar problems but I couldn't find a guide to fix it. Up on further investigation it appeared to come down to the FPS and I created a fix using Nvidia Control Panel.

Please note I only know how to resolve this problem with Nvidia Graphics Cards.

To fix the input lag on Sonic 4 Episode II do the following steps.

1. Open Nvidia Control Panel
2. Select "Manage 3D Settings"
3. Choose the "Program Settings" tab
4. Add Sonic 4 Episode II
5. Change The Following
- Max Frame Rate - 81 FPS
- Preferred Frame Rate - Application-Controlled
- Triple Buffering - On
- Vertical Sync - On
6. Next Click Apply

Now close and relaunch the game if you have it open and the game should now have no input lag. If this doesn't work try changing the Max Frame Rate setting to adjust the games speed. However these settings fixed my game and I was able to play without any issues after this.

Hope it helps!