Here is the answer to the challenge in the previous post.

First /@
 Last[
  SortBy[
    GatherBy[{#, StringJoin[Sort[ToUpperCase[Characters[#]]]]} & 
        /@ DictionaryLookup[{"English", All}],
     Last],
    Length]
]

The idea here is to extract every word from the English dictionary. Then we create pairs consisting of the initial word and the initial word with letters sorted and converted to uppercase. The idea is that the second entry is a anagram key — all words that are anagrams of each other will share the same key.

Then we use GatherBy to consolidate all the pairs with common keys and SortBy to order by the number of anagrams in the set. Finally Last is used to find the words that form the largest anagram set in the English dictionary and First to extract only the words (I drop the keys).

Here is some output that will make the steps clearer.

In[59]:= Take[SortBy[
GatherBy[{#, StringJoin[Sort[ToUpperCase[Characters[#]]]]} & /@
DictionaryLookup[{"English", All}], Last], Length], -10]
Out[59]= {{{“ales”, “AELS”}, {“Elsa”, “AELS”}, {“lase”, “AELS”}, {“leas”,
“AELS”}, {“Lesa”, “AELS”}, {“sale”, “AELS”}, {“seal”, “AELS”}}, {{“capers”,
“ACEPRS”}, {“crapes”, “ACEPRS”}, {“pacers”, “ACEPRS”}, {“parsec”,
“ACEPRS”}, {“recaps”, “ACEPRS”}, {“scrape”, “ACEPRS”}, {“spacer”,
“ACEPRS”}}, {{“carets”, “ACERST”}, {“caster”, “ACERST”}, {“caters”,
“ACERST”}, {“crates”, “ACERST”}, {“reacts”, “ACERST”}, {“recast”,
“ACERST”}, {“traces”, “ACERST”}}, {{“elan”, “AELN”}, {“lane”,
“AELN”}, {“Lane”, “AELN”}, {“lean”, “AELN”}, {“Lean”, “AELN”}, {“Lena”,
“AELN”}, {“Neal”, “AELN”}}, {{“Ingres”, “EGINRS”}, {“reigns”,
“EGINRS”}, {“resign”, “EGINRS”}, {“sering”, “EGINRS”}, {“signer”,
“EGINRS”}, {“singer”, “EGINRS”}, {“Singer”, “EGINRS”}}, {{“notes”,
“ENOST”}, {“onset”, “ENOST”}, {“Seton”, “ENOST”}, {“steno”,
“ENOST”}, {“stone”, “ENOST”}, {“Stone”, “ENOST”}, {“tones”,
“ENOST”}}, {{“opts”, “OPST”}, {“post”, “OPST”}, {“Post”, “OPST”}, {“pots”,
“OPST”}, {“spot”, “OPST”}, {“stop”, “OPST”}, {“tops”, “OPST”}}, {{“pares”,
“AEPRS”}, {“parse”, “AEPRS”}, {“pears”, “AEPRS”}, {“rapes”,
“AEPRS”}, {“reaps”, “AEPRS”}, {“spare”, “AEPRS”}, {“spear”,
“AEPRS”}}, {{“ates”, “AEST”}, {“east”, “AEST”}, {“East”, “AEST”}, {“eats”,
“AEST”}, {“etas”, “AEST”}, {“sate”, “AEST”}, {“seat”, “AEST”}, {“teas”,
“AEST”}}, {{“least”, “AELST”}, {“slate”, “AELST”}, {“Stael”,
“AELST”}, {“stale”, “AELST”}, {“steal”, “AELST”}, {“tales”,
“AELST”}, {“teals”, “AELST”}, {“Tesla”, “AELST”}}}

The above shows the last 10 anagram families with the associated anagram key. The last one is the one used to generate the challenge output.