Wolfram Language (Mathematica), 53 bytes

#/.{a___,Shortest@b___,c___}/;PalindromeQ[a<>c]:>{b}& 

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Input and output are lists of characters. The test suite does the conversion from and to actual strings for convenience.

Wolfram Language (Mathematica), 53 51 bytes

Byte count assumes CP-1252 encoding.

±{a___,Shortest@b___,c___}/;PalindromeQ[a<>c]:={b} 

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Defines a unary operator ± (or a function PlusMinus). Input and output are lists of characters. The test suite does the conversion from and to actual strings for convenience.