Arthur Cayley's first paper on abstract groups, in 1854, can be found in his Collected Papers on the Internet Archive, starting at https://archive.org/stream/collectedmathema02cayluoft#page/122/mode/2up. He states on the 3rd page that groups of prime order are cyclic and then goes on to classify the groups of order 4 and 6 up to isomorphism (he doesn't use that term and refers to the order of an element as its "index," but to modern eyes it is clear what he is doing). If you go to the 5th page of the paper, at https://archive.org/stream/collectedmathema02cayluoft#page/126/mode/2up, he is classifying groups of order 6 and writes near the bottom "we have thus two, and only two, essentially distinct forms of a group of six [symbols]." The next page shows the Cayley tables for these groups, corresponding to the cyclic group and $S_3$, respectively. So far so good.
Moving ahead to 1878, in the first volume of the American Journal of Mathematics Cayley has a short paper on groups. See https://www.jstor.org/stable/2369433?seq=1#page_scan_tab_contents. On the bottom of the second page he writes "The general problem is to find all the groups of a given order $n$." He goes on to say that when $n = 2$ and $3$ there is one group, when $n = 4$ there are two groups (a footnote points out when $n = 5$ there is one group), and then comes the big surprise: "when $n = 6$, there are three groups". That is not only wrong, but contradicts what he wrote in 1854.
Cayley describes the three groups of order 6 as $1, \alpha, \alpha^2, \alpha^3, \alpha^4, \alpha^5$ where $\alpha$ has order 6; and two groups of the form $1, \beta, \beta^2, \alpha, \alpha\beta, \alpha\beta^2$ where $\alpha$ has order 2 and $\beta$ has order 3, distinguished from each other by the rules $\alpha\beta = \beta\alpha$ or $\alpha\beta = \beta^2\alpha$. (Strictly speaking, he lists that last relation together with $\alpha\beta^2 = \beta\alpha$, but it's redundant since it follows from $\alpha\beta = \beta^2\alpha$ since $\beta^3 = 1$.) The second group in his list is recognizable as a direct product of a group of order 2 and a group of order 3, so it appears that Cayley does not realize his second group is isomorphic to his first group. The mistake is understandable; even today, students learning group theory who do not yet know the Chinese remainder theorem could easily believe at first that a direct product of cyclic groups of order 2 and 3 is not isomorphic to a cyclic group of order 6 since the groups don't "look" the same (confusing appearance with structure).
My question is this: did Cayley in print later acknowledge his mistake or explain why he got it right in 1854 and later got it wrong? In a 1913 Amer. Math. Monthly paper G. A. Miller writes about Cayley's mistake on https://www.jstor.org/stable/2973510?seq=4#page_scan_tab_contents, but is silent about Cayley ever issuing a correction or explanation later.
Note: Near the end of writing up this question I found Cayley's error mentioned by Richard Stanley in a Mathoverflow post about mathematical errors -- see https://mathoverflow.net/questions/879/most-interesting-mathematics-mistake/921 -- but my question still stands: did Cayley ever bring up his error after 1878?
