The book of the Acts of Solomon, also known as the book of the Annals of Solomon, was a written source of information about Solomon’s life and administration, which was used by the writer of what Christians list in the Hebrew scriptures as 1 and 2 Kings. The book of the Annals of Solomon is mentioned in 1 Kings 11:4. However, the author of 1 and 2 Kings used a variety of sources in compiling the history of the monarchy. For example:
The book of the Annals of the kings of Israel: A record of the reigns of the kings of the northern kingdom (not to be confused with the canonical book of 1 and 2 Chronicles, which was written later than 1 and 2 Kings, and contains the history of the kings of Judah only). 1 Kings 14:19: "The other events of Jeroboam’s reign, his wars and how he ruled, are written in the book of the Annals of the kings of Israel."
The book of the Annals of the kings of Judah: A record of the reigns of the kings of Judah, similar to that of the kings of the northern kingdoms. 1 Kings 14:29: "As for the other events of Rehoboam’s reign, and all he did, are they not written in the book of the annals of the kings of Judah?" See also 1 Kings 15:7 and 23.
There are many references to other writings, e.g., the records of Samuel the seer, the records of Nathan the prophet, and the records of Gad the seer (1 Chronicles 29:29), the events of Uzziah’s reign (2 Chronicles 26:22). There is no telling if they still exist, yet we have all that we need in the inspired books of the Old Testament.
Solomon is traditionally attributed as the author of the Song of Solomon (or Song of Songs) as suggested by the opening line and references to his name throughout the book. In 1 Kings 4:32, King Solomon’s body of writing is mentioned: “He composed some 3,000 proverbs and wrote 1,005 songs.”
Regarding the book of the Acts of Solomon mentioned in 1 Kings 11:41, we do not know where this book is and what else Solomon did according to it.
A book called ‘The Wisdom of Solomon' is based on an ancient manuscript known as the Muratorian fragment which says it was written by “the friends of Solomon in his honor.” It is widely accepted that Solomon did not write the book, which dates back to the 1st or 2nd century B.C., centuries after his death.
https://www.gotquestions.org/Wisdom-of-Solomon.html
Nearly 1,500 years after the death of Solomon, another book appeared called the ‘Testament of Solomon’, claiming to record his supernatural adventures, e.g. Solomon’s enslavement of demons and his building of the temple through the power of a magic ring. The stories in the Testament of Solomon intermingle Greek, Egyptian, and Christian spiritual ideas. A type of medical alchemy is also in this writing. Demons speaking with Solomon take the blame for certain ailments and relate spells that can be used to remove their power. This text does not agree with inspired Scripture and were not accepted in any sense by the early church. Jewish scholars considered it fictional, as well. Source: https://www.gotquestions.org/Testament-of-Solomon.html
Answer: We do not know what happened to 'the book of the Acts of Solomon'. However, we know that the book ‘the Wisdom of Solomon’ and ‘the Testament of Solomon’ are both rejected as uninspired by God, are contradictory to Bible truth and theology, as well as being written long after the death of Solomon.