fälschlicherweise
mistakenly
Based on an error or a false assumption; wrongfully.
Er wurde fälschlicherweise des Diebstahls beschuldigt.
He was mistakenly accused of theft.

mistakenly
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mistakenly
Based on an error or a false assumption; wrongfully.
Er wurde fälschlicherweise des Diebstahls beschuldigt.
He was mistakenly accused of theft.

The word is a typical adverb formed with the ending '-weise'. The base is the adjective 'fälschlich' (false, erroneous), which is derived from the word 'falsch' (wrong).
Die Zeitung hat fälschlicherweise einen falschen Namen gedruckt.
The newspaper mistakenly printed the wrong name.
As an adverb, 'fälschlicherweise' (mistakenly) is flexible in sentence position. It can stand at the beginning of the sentence (position 1), in the middle of the sentence, or after the conjugated verb.
Fälschlicherweise dachte ich, das Treffen sei erst morgen.
Mistakenly, I thought the meeting was not until tomorrow.
'Fälschlicherweise' (mistakenly) is often used with verbs of thinking, saying, or assuming. Typical combinations are 'fälschlicherweise annehmen' (to mistakenly assume), 'fälschlicherweise glauben' (to mistakenly believe), or 'fälschlicherweise behaupten' (to mistakenly claim).
Viele Leute nehmen fälschlicherweise an, dass dieses Gesetz bereits gilt.
Many people mistakenly assume that this law is already in effect.
accidentally
'versehentlich' and 'fälschlicherweise' can both be used with mistakes, but they emphasize different things. 'versehentlich' means that an action happens by accident, without intention. 'fälschlicherweise' means that something is based on a false assumption or happens wrongfully. So it is more about an incorrect judgment, piece of information, or decision. Learners may confuse the words because in both cases something is not right. 'versehentlich' fits a mishap; 'fälschlicherweise' fits an error or a false assignment.

Ich wurde fälschlicherweise als Gast auf der Liste eingetragen.
I was mistakenly entered on the list as a guest.
Tom löscht versehentlich die Nachricht, bevor er sie lesen kann.
Tom accidentally deletes the message before he can read it.