When combined with the conjunction しかし ‘however’, the particle ながら ‘while’, or ‘although’ creates a new word which itself is used as a conjunction, or occasionally adverbially.
しかしながら is itself often translated as ‘however’, or ‘nevertheless’, although it is stronger than しかし by itself.
A more literal translation which may carry the strength of this grammar pattern a bit better would be ‘while however, (B)’, or ‘even while true, (B)’. Just like in English, the extension of the phrase in Japanese adds emphasis to the overall meaning.
しかしながら will primarily appear at the beginning of a new sentence when referring back to something that has previously been stated, either by the speaker themselves or by another.
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日本（にほん）は安全（あんぜん）な国（くに）だと言（い）われている。しかしながら100％安全（あんぜん）というわけでもない。
Japan is said to be a safe country. However, it is not 100% safe.
精一杯（せいいっぱい）頑張（がんば）れば夢（ゆめ）が叶（かな）うと言（い）われている。しかしながら、人生（じんせい）はそう甘（あま）くない。
It is said that if you work as hard as you can, your dreams will come true. However, life is not so easy.
そのアイデアはいいと思（おも）います。しかしながら、我々（われわれ）の予算（よさん）だとそのプランを実行（じっこう）することはできないでしょう。
I think the idea is good. However, our budget will not allow us to implement that plan.
As
しかしながら is primarily used as an emphasized version of しかし, which itself is formal,
しかしながら will primarily be seen in literature, or in formal spoken language.