From quick testing, this works (doesn't support the div tags for HTML4). The good part is that the "summary" Special Block is automatically handled by ox-html.
((string= block-type "details")
;; Recognize Org Special blocks like:
;; #+begin_details
;; #+begin_summary
;; This is summary.
;; #+end_summary
;; Here are the details.
;; #+end_details
(let* ((is-open (member "open" (org-element-property :attr_html special-block)))
(str1 (org-blackfriday-special-block special-block contents nil))
;; Insert a new-line before the closing </details> tag
;; for correct Markdown parsing for cases when the
;; Special Block content ends a code block. Without this
;; inserted newline, the Markdown converted content will
;; look like below, and Blackfriday won't parse it
;; correctly.
;; ```emacs-lisp
;; (message "a code block")
;; ```</details>
;; A closing </p> tag is also added.. the opening <p>
;; tag is later added in the `str2' var if summary is
;; present, else in `str3' var.
(str1 (replace-regexp-in-string "</details>\\'" "\n</p>\\&" str1))
;; Detect the summary closing tag "</summary>".
;; Also add the opening <p> tag with "details" class
;; so that just as CSS rules can be set for summary
;; ("details summary"), they can be set for the details
;; portion following the <summary> too, using "details
;; .details".
(str2 (replace-regexp-in-string
"\\(<summary>\\(.\\|\n\\)*</summary>\\)"
"\\1<p class=\"details\">"
str1)))
;; (message "[DBG details/summary]: is-open:%S `%s' `%s'" is-open str1 str2)
(if is-open
(replace-regexp-in-string "\\`\\(<details\\)>" "\\1 open>" str2)
str2)))
Also, thinking about HTML4 support, as details and summary tags were added in HTML5, the "support" would be simply to and the (string= block-type "details") condition with org-html-html5-fancy or equivalent. So then for HTML4, details and summary Special Blocks will simply be div tags.