Keeping metadata (time/date) when extracting frames from video Circumstances

I recently started to extract frames from videos (GoPro) for underwater photography. I'm currently using Adobe Premiere Pro to do so, which works, but when I imported these new photos into Lightroom, I realized that the date/time metadata of the pictures don't match with the video's metadata (they have the date and time of when I extracted them, for obvious reasons).

This is really not convenient, since I need to manually re-adjust the time for each of these new pictures (I want all of my pictures to have accurate date/time metadata, since I often look back and want to know when I took these pictures).

Question

How can I extract a frame from a movie file and store the date/time metadata from the video to the frame? If this is possible, how can I store the time difference between the location of the frame and the start of the video?

Thank you very much.
If you are saving frames from PP using the snapshot tool, then by default it's saving a file that's of the form SEQUENCENAME_TIMECODE.  If you know the date/time of the start of the sequence, you can compute frame-accurate metadata by adding the timecode of the image to the adjusted datetime of the sequence start.  (By adjusted datetime, I mean that if your sequence starts at 01:00:00:00 and your actual clip was 16:20:25:15 on Apr 1, 2014, then you need to subtract one hour from the clip time to account for the SMPTE offset).

As for programming languages that will do what you want, look at bash (the Bourne Again Shell), Python, and/or Perl.  Perl is wicked cool in how it can slice and dice strings, do math, and put the strings back together.  But it is also wicked hard to learn idiomatically.  Python may be the best starting point.

EXIFtool is a great way to embed correct metadata into image files: http://www.sno.phy.queensu.ca/~phil/exiftool/

And it's a way to learn some Perl while you are at it.