Correct file extensions

You can do it relatively easily in bash:

for f in *jpg; do 
    type=$(file -0 -F" " "$f" | grep -aPo '\0\s*\K\S+') 
    mv "$f" "${f%%.*}.${type,,}"  
done

This is the same idea as @A.B's answer but using shell globs instead of find. The ${f%%.*} is the filename without its extension. The -0 of the file command makes it print a \0 after the file name which we then use to grep the file type. This should work with arbitrary file names, including those that contain spaces, newlines or anything else. The ${type,,} is a trick to get lower case extensions. It would convert PNG to png.

You didn't say in your question, but if you need this to be recursive and descend into subdirectories, you could use this instead:

shopt -s globstar
for f in **/*jpg; do 
    type=$(file -0 -F" " "$f" | grep -aPo '\0\s*\K\S+') 
    mv "$f" "${f%%.*}.${type,,}"  
done

The shopt -s globstar will enable bash's globstar option which lets ** match subdirectories:

globstar

If set, the pattern ** used in a pathname expansion context will match all files and zero or more directories and subdirectories. If the pattern is followed by a /, only directories and subdirectories match.

The script below can be used to (recursively) rename an incorrectly set extension, .jpg, to the correct one. In case it finds an unreadable file, it will report it in the script's output.

The script use the imghdr module, to recognize the following types: rgb, gif, pbm, pgm, ppm, tiff, rast, xbm, jpeg, bmp, png. More on the imghdr module here. The list can be extended with more types, as mentioned in the link.

As it is, it specifically renames files with the extension .jpg, as mentioned in the question. With a minor change, it can be fit to rename any extension, or a specific set of extensions, into the correct one (or with no extension, like here).

The script:

#!/usr/bin/env python3
import os
import imghdr
import shutil
import sys

directory = sys.argv[1]

for root, dirs, files in os.walk(directory):
    for name in files:
        file = root+"/"+name
        # find files with the (incorrect) extension to rename
        if name.endswith(".jpg"):
            # find the correct extension
            ftype = imghdr.what(file)
            # rename the file
            if ftype != None:
                shutil.move(file, file.replace("jpg",ftype))
            # in case it can't be determined, mention it in the output
            else:
                print("could not determine: "+file)

How to use

1. Copy the script into an empty file, save it as rename.py

2. Run it by the command:

   python3 /path/to/rename.py <directory>
   

Note: My approach seems to be too complex. I would prefer terdons answer in your place.

You can use the command file to to determine the file type:

% file 20050101_14-24-37_330.jpg 
20050101_14-24-37_330.jpg: JPEG image data, EXIF standard 2.2, baseline, precision 8, 1200x1600, frames 3

% file test.jpg
test.jpg: PNG image data, 1192 x 774, 8-bit/color RGBA, non-interlaced

With this information, the files can be renamed:

Please do a test before you apply the command to your images

find . -type f -iname "*.jpg" -print0 | xargs -0 -I{} file -F"<separator>" {} | 
 awk -F " image data" '{print $1}' | 
  awk -F"<separator> " '{
   system("mv \""$1"\" $(dirname \""$1"\")/$(basename -s .jpg \"" $1 "\")."$2)
   }'

Example

% find . -type f -name "*.jpg"
./test.jpg
./sub/20050101_14-24-37_330.jpg

% find . -type f -iname "*.jpg" -print0 | xargs -0 -I{} file -F"<separator>" {} | awk -F " image data" '{print $1}' | awk -F"<separator> " '{system ("mv \""$1"\" $(dirname \""$1"\")/$(basename -s .jpg \"" $1 "\")."$2)}'

% find . -type f -iname "*"    
./test.PNG
./sub/20050101_14-24-37_330.JPEG