Augusta was a princess of Schleswig-Holstein and not saxe-weimar. The Emperor wilhem did not have a brother named Karl only Henry. His other two brothers sigismund and waldemar died in childhood.
Princess Augusta of Saxe-Weimar-Eisenach (1811-1890) was the second daughter of Grand Duke Carl Friedrich of Saxe-Weimar-Eisenach (1783-1863) and Grand Duchess Maria Pavlovna of Russia (1786-1859). She married the future Kaiser Wilhelm I and was the mother of Friedrich III and mother-in-law of Victoria of Great Britain.
Augusta's siblings, Grand Duke Carl Alexander of Saxe-Weimar-Eisenach (1818-1901) and Princess Maria of Saxe-Weimar-Eisenach (1808-1877) both mad interesting marriages.
Grand Duke Carl Alexander married his first cousin Princess Sophia of the Netherlands (1824-1897), daughter of King Willem II of the Netherlands and of Grand Duchess Anna Pavlovna of Russia (1795-1865), a younger sister of Maria Pavlovna.
Princess Maria of Saxe-Weimar-Eisenach married Prince Karl of Prussia (1801-1883), a younger brother of Kaiser Wilhelm I. Furthermore, Wilhelm and Karl's sister, Princess Charlotte of Prussia, better known in Russian history as Tsaritsa Alexandra Feodorovna, was the consort of none other than Nicholas I, the brother of both Anna and Maria Pavlovna.
Arturo Beéche
PS: Sources: THE GRAND DUCHESSES (EUROHISTORY 2004), page 32 & Burke's Royal Families of the World, page 261.