1. Find mass of carbon from $\mathrm{CO_2}$

From combustion, $44\text{ g of }\mathrm{CO_2} \text{ contains } 12\text{ g of C}$

So, carbon in $0.220\,$g of $\mathrm{CO_2}$ is $m_C = 0.220\times \frac{12}{44} = 0.060\text{ g}$

2. Use given percentage of carbon to find mass of compound

Given that carbon percentage is $24\%$, $\frac{0.060}{m_{\text{compound}}}\times 100 = 24$

Hence, $m_{\text{compound}} = \frac{0.060\times 100}{24} = 0.25\text{ g}$

3. Find mass of hydrogen from $\mathrm{H_2O}$

From combustion, $18\text{ g of }\mathrm{H_2O} \text{ contains } 2\text{ g of H}$

So, hydrogen in $0.126\,$g of $\mathrm{H_2O}$ is $m_H = 0.126\times \frac{2}{18} = 0.014\text{ g}$

4. Calculate percentage of hydrogen

$\%H = \frac{0.014}{0.25}\times 100 = 5.6\%$

Now the question asks for: $\%\text{ of hydrogen} = \underline{\hspace{2cm}}\times 10^{-1}$

Since $5.6 = 56\times 10^{-1}$

the required nearest integer is $\boxed{56}$

5. Comparison with stored answer

Stored correct answer = $56$

This matches our derived answer.