Is there a term for a word inside another word?

Is there a term for a word that occurs unbroken within another word? For example, the word "fun" in "funeral", or "drag" in "hydragog".

The closest thing I could find from my search was the term "kangaroo word", which refers to a word that contains the letters of its synonym, in the correct order (though not necessarily consecutively).

I am currently using "substring" as a substitute in my essay. Unfortunately, "substring" applies to any selection of consecutive characters from the containing word, and doesn't convey the fact that the subset is a complete English word.

I have tagged this phrase-requests, but I would appreciate single words as well. I will have to use the term quite often, so the less awkward it is, the better.

There is a historical "question" on ELU about Matryoshka words (matryoshka is a Russian doll which contains identical smaller dolls) which might provide a term to use, although that would be easier to apply to the long word rather than the substring.

It seems that kangaroo word is rather apt, but it's extremely limiting to apply it to a word containing self-synonyms. I see no reason why you should not use joey word to describe the short word which is included in the longer one. You might need to explain why you have chosen "joey" as it may not be obvious that it's the term for a baby kangaroo; and if you use it as an ordinary noun you may need to set it off in italics to indicate its technical use:

The word hydragog contains the joey words Hydra, drag and agog.